C. DUMITRESCU     V. SELEACU

                                    editors
                             Department of Mathematics
                               University of Craiova
                     





                          SOME NOTIONS AND QUESTIONS
                                      IN
                                NUMBER THEORY

                               (fourth edition)



                             American Research Press
                                
                                      1998
                                       



        Introduction




    The following notions, definitions, unsolved problems, questions, 
  theorems, corollaries, formulae, conjectures, examples, mathematical 
  criteria, etc. ( on integer sequences, numbers, quotients, residues, 
  exponents, sieves, pseudo-primes/squares/cubes/factorials, almost primes, 
  mobile periodicals, functions, tables, prime/square/factorial bases, etc. )
  have been extracted from the Archives of American Mathematics (University 
  of Texas at Austin), Arizona State University (Tempe): "The Florentin 
  Smarandache papers" special collection, University of Craiova Library, 
  and Arhivele Statului (Filiala Valcea).


    Any comments, solutions, generalizations, cross-reference problems, 
  applications, etc. concerning this material are welcome and if correct they 
  will be published on our journal.
  Articles, notes, solved or unsolved proposed problems, etc. should be sent
  on the following address:
                            Dr. L. M. Perez, Editor
                            American Research Press
                            Rehoboth, Box 141
                            NM 87322, USA
                            E-mail:  ARP@cia-g.com
                            URL:  http://www.gallup.unm.edu/~smarandache/

    All contributions will be published, and the authors will get free copies
  of the <Smarandache Notions Journal> with their published paper. 
  The journal is worldwide distributed: in electronic version and hard copy.
  It reaches more than 5,000 readers around the world.  
    We prefer electronic papers (by e-mail, or diskets sent by regular mail),
  but hard copies (camera ready) are also welcome.
    We prefer files in Microsoft Word 7.0a or lower for Windows 95, 
  or Word Perfect 6.1 or lower, or text files.
    The best three papers will be awarded with $300.00 each.

  The editors are also looking for long lenght manuscripts dealing with the
  Smarandache type notions: at least 100 pages, camera ready.  The manuscripts
  will be published under the form of a book, and the authors will receive
  100 free copies as copyright.

  Any letter will be answered.









                                                              The Editors







  1)Smarandache consecutive sequence:
      1,12,123,1234,12345,123456,1234567,12345678,123456789,12345678910, 
      1234567891011,123456789101112,12345678910111213,...
        Florentin Smarandache asked how many primes are there among 
      these numbers?
        In a general form, the Smarandache Consecutive Sequence is considered
      in an arbitrary numeration base B.

     References:
      Student Conference, University of Craiova, Department of Mathematics,
        April 1979,  "Some problems in number theory" by Florentin Smarandache.
      Arizona State University, Hayden Library, "The Florentin Smarandache
        papers" special collection, Tempe, AZ 85287-1006, USA, phone: 
        (602)965-6515 (Carol Moore librarian), email: ICCLM@ASUACAD.BITNET . 
      "The Encyclopedia of Integer Sequences", by N. J. A. Sloane and 
        S. Plouffe, Academic Press, San Diego, New York, Boston, London, 
        Sydney, Tokyo, Toronto, 1995;
        also online, email:  superseeker@research.att.com ( SUPERSEEKER by 
        N. J. A. Sloane, S. Plouffe, B. Salvy,  ATT Bell Labs, Murray Hill,
        NJ 07974, USA); 
      N. J. A. Sloane, e-mails to R. Muller, February 13 - March 7, 1995.



  2)Smarandache circular sequence:
        1,12,21,123,231,312,1234,2341,3412,4123,12345,23451,34512,45123,51234,
        | |   | |         | |                 | |                           |
           ---   ---------   -----------------   ---------------------------
        1   2        3               4                         5

    
        123456,234561,345612,456123,561234,612345,1234567,2345671,3456712,...
        |                                       | |
         ---------------------------------------   ---------------------- ...
                        6                                          7


  3)Smarandache symmetric sequence:
       1,11,121,1221,12321,123321,1234321,12344321,123454321,1234554321,
       12345654321,123456654321,1234567654321,12345677654321,123456787654321,
       1234567887654321,12345678987654321,123456789987654321,
       12345678910987654321,1234567891010987654321,123456789101110987654321,
       12345678910111110987654321,...
        Florentin Smarandache asked how many primes are there among
      these numbers?
        In a general form, the Smarandache Symmetric Sequence is considered
      in an arbitrary numeration base B.

     References:
      Student Conference, University of Craiova, Department of Mathematics,
        April 1979,  "Some problems in number theory" by Florentin Smarandache.
      Arizona State University, Hayden Library, "The Florentin Smarandache
        papers" special collection, Tempe, AZ 85287-1006, USA, phone: 
        (602)965-6515 (Carol Moore librarian), email: ICCLM@ASUACAD.BITNET . 
      "The Encyclopedia of Integer Sequences", by N. J. A. Sloane and 
        S. Plouffe, Academic Press, San Diego, New York, Boston, London, 
        Sydney, Tokyo, Toronto, 1995;
        also online, email:  superseeker@research.att.com ( SUPERSEEKER by 
        N. J. A. Sloane, S. Plouffe, B. Salvy,  ATT Bell Labs, Murray Hill,
        NJ 07974, USA); 

        
  4)Smarandache deconstructive sequence: 
      1,23,456,7891,23456,789123,4567891,23456789,123456789,1234567891, ...
      |          ||         ||        ||        | |       | |       ||        
       ----------  ---------  --------  --------   --------  -------  - ...


     References:
      Florentin Smarandache, "Only Problems, not Solutions!", Xiquan 
        Publishing House, Phoenix-Chicago, 1990, 1991, 1993; 
        ISBN: 1-879585-00-6.
        (reviewed in <Zentralblatt fur Mathematik> by P. Kiss: 11002, 
         pre744, 1992;
         and <The American Mathematical Monthly>, Aug.-Sept. 1991);
      Arizona State University, Hayden Library, "The Florentin Smarandache
        papers" special collection, Tempe, AZ 85287-1006, USA, phone: 
        (602)965-6515 (Carol Moore librarian), email: ICCLM@ASUACAD.BITNET . 
      "The Encyclopedia of Integer Sequences", by N. J. A. Sloane and 
        S. Plouffe, Academic Press, San Diego, New York, Boston, London, 
        Sydney, Tokyo, Toronto, 1995;
        also online, email:  superseeker@research.att.com ( SUPERSEEKER by 
        N. J. A. Sloane, S. Plouffe, B. Salvy,  ATT Bell Labs, Murray Hill,
        NJ 07974, USA); 



  5)Smarandache mirror sequence:
       1,212,32123,4321234,543212345,65432123456,7654321234567,876543212345678,
       98765432123456789,109876543212345678910,1110987654321234567891011,...

     Question:  How many of them are primes?


  6)Smarandache permutation sequence:
       12,1342,135642,13578642,13579108642,135791112108642,1357911131412108642,
       13579111315161412108642,135791113151718161412108642, 
       1357911131517192018161412108642,...

     Question:  Is there any perfect power among these numbers?
     (Their last digit should be: 
      either 2 for exponents of the form 4k+1, 
      either 8 for exponents of the form 4k+3, where k >= 0 .)

     Smarandache conjectures: no!


  7)Smarandache generalizated permutation sequence:
      If g(n), as a function, gives the number of digits of a(n), and F if a
      permutation of g(n) elements, then:
             __________________
      a(n) = F(1)F(2)...F(g(n)) .
                                               

  8)Smarandache mobile periodicals (I):
   ...000000000000000000000000000000010000000000000000000000000000000000...
   ...000000000000000000000000000000111000000000000000000000000000000000...
   ...000000000000000000000000000001101100000000000000000000000000000000...
   ...000000000000000000000000000000111000000000000000000000000000000000...
   ...000000000000000000000000000000010000000000000000000000000000000000...
   ...000000000000000000000000000000111000000000000000000000000000000000...
   ...000000000000000000000000000001101100000000000000000000000000000000...
   ...000000000000000000000000000011000110000000000000000000000000000000...
   ...000000000000000000000000000001101100000000000000000000000000000000...
   ...000000000000000000000000000000111000000000000000000000000000000000...
   ...000000000000000000000000000000010000000000000000000000000000000000...
   ...000000000000000000000000000000111000000000000000000000000000000000...
   ...000000000000000000000000000001101100000000000000000000000000000000...
   ...000000000000000000000000000011000110000000000000000000000000000000...
   ...000000000000000000000000000110000011000000000000000000000000000000...
   ...000000000000000000000000000011000110000000000000000000000000000000...
   ...000000000000000000000000000001101100000000000000000000000000000000...
   ...000000000000000000000000000000111000000000000000000000000000000000...
   ...000000000000000000000000000000010000000000000000000000000000000000...
   ...000000000000000000000000000000111000000000000000000000000000000000...
   ...000000000000000000000000000001101100000000000000000000000000000000...
   ...000000000000000000000000000011000110000000000000000000000000000000...
   ...000000000000000000000000000110000011000000000000000000000000000000...
   ...000000000000000000000000001100000001100000000000000000000000000000...
   ...000000000000000000000000000110000011000000000000000000000000000000...
   ...000000000000000000000000000011000110000000000000000000000000000000...
   ...000000000000000000000000000001101100000000000000000000000000000000...
   ...000000000000000000000000000000111000000000000000000000000000000000...
   ...000000000000000000000000000000010000000000000000000000000000000000...
   ...000000000000000000000000000000111000000000000000000000000000000000...
   ...000000000000000000000000000001101100000000000000000000000000000000...
   ...000000000000000000000000000011000110000000000000000000000000000000...
   ...000000000000000000000000000110000011000000000000000000000000000000...
   ...000000000000000000000000001100000001100000000000000000000000000000...
   ...000000000000000000000000011000000000110000000000000000000000000000...
   ...000000000000000000000000001100000001100000000000000000000000000000...
   ...000000000000000000000000000110000011000000000000000000000000000000...
   ...000000000000000000000000000011000110000000000000000000000000000000...
   ...000000000000000000000000000001101100000000000000000000000000000000...
   ...000000000000000000000000000000111000000000000000000000000000000000...
   ...000000000000000000000000000000010000000000000000000000000000000000...
   ...000000000000000000000000000000111000000000000000000000000000000000...
   ...000000000000000000000000000001101100000000000000000000000000000000...
   ...000000000000000000000000000011000110000000000000000000000000000000...
   ...000000000000000000000000000110000011000000000000000000000000000000...
   ...000000000000000000000000001100000001100000000000000000000000000000...
   ...000000000000000000000000011000000000110000000000000000000000000000...
   ...000000000000000000000000110000000000011000000000000000000000000000...
   ........................................................................



   This sequence has the form
 1,111,11011,111,1,111,11011,1100011,11011,111,1,111,11011,1100011,110000011,...
 |               |                             |
  ---------------|                             |
         5       |                             |
                  -----------------------------|
                                 7             |
                                                ---------------------------- ...
                                                                      9


  9)Smarandache mobile periodicals (II):
   ...000000000000000000000000000000010000000000000000000000000000000000...
   ...000000000000000000000000000000111000000000000000000000000000000000...
   ...000000000000000000000000000001121100000000000000000000000000000000...
   ...000000000000000000000000000000111000000000000000000000000000000000...
   ...000000000000000000000000000000010000000000000000000000000000000000...
   ...000000000000000000000000000000111000000000000000000000000000000000...
   ...000000000000000000000000000001121100000000000000000000000000000000...
   ...000000000000000000000000000011232110000000000000000000000000000000...
   ...000000000000000000000000000001121100000000000000000000000000000000...
   ...000000000000000000000000000000111000000000000000000000000000000000...
   ...000000000000000000000000000000010000000000000000000000000000000000...
   ...000000000000000000000000000000111000000000000000000000000000000000...
   ...000000000000000000000000000001121100000000000000000000000000000000...
   ...000000000000000000000000000011232110000000000000000000000000000000...
   ...000000000000000000000000000112343211000000000000000000000000000000...
   ...000000000000000000000000000011232110000000000000000000000000000000...
   ...000000000000000000000000000001121100000000000000000000000000000000...
   ...000000000000000000000000000000111000000000000000000000000000000000...
   ...000000000000000000000000000000010000000000000000000000000000000000...
   ...000000000000000000000000000000111000000000000000000000000000000000...
   ...000000000000000000000000000001121100000000000000000000000000000000...
   ...000000000000000000000000000011232110000000000000000000000000000000...
   ...000000000000000000000000000112343211000000000000000000000000000000...
   ...000000000000000000000000001123454321100000000000000000000000000000...
   ...000000000000000000000000000112343211000000000000000000000000000000...
   ...000000000000000000000000000011232110000000000000000000000000000000...
   ...000000000000000000000000000001121100000000000000000000000000000000...
   ...000000000000000000000000000000111000000000000000000000000000000000...
   ...000000000000000000000000000000010000000000000000000000000000000000...
   ...000000000000000000000000000000111000000000000000000000000000000000...
   ...000000000000000000000000000001121100000000000000000000000000000000...
   ...000000000000000000000000000011232110000000000000000000000000000000...
   ...000000000000000000000000000112343211000000000000000000000000000000...
   ...000000000000000000000000001123454321100000000000000000000000000000...
   ...000000000000000000000000011234565432110000000000000000000000000000...
   ...000000000000000000000000001123454321100000000000000000000000000000...
   ...000000000000000000000000000112343211000000000000000000000000000000...
   ...000000000000000000000000000011232110000000000000000000000000000000...
   ...000000000000000000000000000001121100000000000000000000000000000000...
   ...000000000000000000000000000000111000000000000000000000000000000000...
   ...000000000000000000000000000000010000000000000000000000000000000000...
   ...000000000000000000000000000000111000000000000000000000000000000000...
   ...000000000000000000000000000001121100000000000000000000000000000000...
   ...000000000000000000000000000011232110000000000000000000000000000000...
   ...000000000000000000000000000112343211000000000000000000000000000000...
   ...000000000000000000000000001123454321100000000000000000000000000000...
   ...000000000000000000000000011234565432110000000000000000000000000000...
   ...000000000000000000000000112345676543211000000000000000000000000000...
   ........................................................................



   This sequence has the form
 1,111,11211,111,1,111,11211,1123211,11211,111,1,111,11211,1123211,112343211,...
 |               |                             |
  ---------------|                             |
         5       |                             |
                  -----------------------------|
                                 7             |
                                                ---------------------------- ...
                                                                      9



  10)Smarandache infinite numbers (I):
   ...111111111111111111111111111111101111111111111111111111111111111111...
   ...111111111111111111111111111111000111111111111111111111111111111111...
   ...111111111111111111111111111110010011111111111111111111111111111111...
   ...111111111111111111111111111111000111111111111111111111111111111111...
   ...111111111111111111111111111111101111111111111111111111111111111111...
   ...111111111111111111111111111111000111111111111111111111111111111111...
   ...111111111111111111111111111110010011111111111111111111111111111111...
   ...111111111111111111111111111100111001111111111111111111111111111111...
   ...111111111111111111111111111110010011111111111111111111111111111111...
   ...111111111111111111111111111111000111111111111111111111111111111111...
   ...111111111111111111111111111111101111111111111111111111111111111111...
   ...111111111111111111111111111111000111111111111111111111111111111111...
   ...111111111111111111111111111110010011111111111111111111111111111111...
   ...111111111111111111111111111100111001111111111111111111111111111111...
   ...111111111111111111111111111001111100111111111111111111111111111111...
   ...111111111111111111111111111100111001111111111111111111111111111111...
   ...111111111111111111111111111110010011111111111111111111111111111111...
   ...111111111111111111111111111111000111111111111111111111111111111111...
   ...111111111111111111111111111111101111111111111111111111111111111111...
   ...111111111111111111111111111111000111111111111111111111111111111111...
   ...111111111111111111111111111110010011111111111111111111111111111111...
   ...111111111111111111111111111100111001111111111111111111111111111111...
   ...111111111111111111111111111001111100111111111111111111111111111111...
   ...111111111111111111111111110011111110011111111111111111111111111111...
   ...111111111111111111111111111001111100111111111111111111111111111111...
   ...111111111111111111111111111100111001111111111111111111111111111111...
   ...111111111111111111111111111110010011111111111111111111111111111111...
   ...111111111111111111111111111111000111111111111111111111111111111111...
   ...111111111111111111111111111111101111111111111111111111111111111111...
   ...111111111111111111111111111111000111111111111111111111111111111111...
   ...111111111111111111111111111110010011111111111111111111111111111111...
   ...111111111111111111111111111100111001111111111111111111111111111111...
   ...111111111111111111111111111001111100111111111111111111111111111111...
   ...111111111111111111111111110011111110011111111111111111111111111111...
   ...111111111111111111111111100111111111001111111111111111111111111111...
   ...111111111111111111111111110011111110011111111111111111111111111111...
   ...111111111111111111111111111001111100111111111111111111111111111111...
   ...111111111111111111111111111100111001111111111111111111111111111111...
   ...111111111111111111111111111110010011111111111111111111111111111111...
   ...111111111111111111111111111111000111111111111111111111111111111111...
   ...111111111111111111111111111111101111111111111111111111111111111111...
   ...111111111111111111111111111111000111111111111111111111111111111111...
   ...111111111111111111111111111110010011111111111111111111111111111111...
   ...111111111111111111111111111100111001111111111111111111111111111111...
   ...111111111111111111111111111001111100111111111111111111111111111111...
   ...111111111111111111111111110011111110011111111111111111111111111111...
   ...111111111111111111111111100111111111001111111111111111111111111111...
   ...111111111111111111111111001111111111100111111111111111111111111111...
   ........................................................................



  11)Smarandache infinite numbers (II): 
   ...111111111111111111111111111111121111111111111111111111111111111111...
   ...111111111111111111111111111111222111111111111111111111111111111111...
   ...111111111111111111111111111112232211111111111111111111111111111111...
   ...111111111111111111111111111111222111111111111111111111111111111111...
   ...111111111111111111111111111111121111111111111111111111111111111111...
   ...111111111111111111111111111111222111111111111111111111111111111111...
   ...111111111111111111111111111112232211111111111111111111111111111111...
   ...111111111111111111111111111122343221111111111111111111111111111111...
   ...111111111111111111111111111112232211111111111111111111111111111111...
   ...111111111111111111111111111111222111111111111111111111111111111111...
   ...111111111111111111111111111111121111111111111111111111111111111111...
   ...111111111111111111111111111111222111111111111111111111111111111111...
   ...111111111111111111111111111112232211111111111111111111111111111111...
   ...111111111111111111111111111122343221111111111111111111111111111111...
   ...111111111111111111111111111223454322111111111111111111111111111111...
   ...111111111111111111111111111122343221111111111111111111111111111111...
   ...111111111111111111111111111112232211111111111111111111111111111111...
   ...111111111111111111111111111111222111111111111111111111111111111111...
   ...111111111111111111111111111111121111111111111111111111111111111111...
   ...111111111111111111111111111111222111111111111111111111111111111111...
   ...111111111111111111111111111112232211111111111111111111111111111111...
   ...111111111111111111111111111122343221111111111111111111111111111111...
   ...111111111111111111111111111223454322111111111111111111111111111111...
   ...111111111111111111111111112234565432211111111111111111111111111111...
   ...111111111111111111111111111223454322111111111111111111111111111111...
   ...111111111111111111111111111122343221111111111111111111111111111111...
   ...111111111111111111111111111112232211111111111111111111111111111111...
   ...111111111111111111111111111111222111111111111111111111111111111111...
   ...111111111111111111111111111111121111111111111111111111111111111111...
   ...111111111111111111111111111111222111111111111111111111111111111111...
   ...111111111111111111111111111112232211111111111111111111111111111111...
   ...111111111111111111111111111122343221111111111111111111111111111111...
   ...111111111111111111111111111223454322111111111111111111111111111111...
   ...111111111111111111111111112234565432211111111111111111111111111111...
   ...111111111111111111111111122345676543221111111111111111111111111111...
   ...111111111111111111111111112234565432211111111111111111111111111111...
   ...111111111111111111111111111223454322111111111111111111111111111111...
   ...111111111111111111111111111122343221111111111111111111111111111111...
   ...111111111111111111111111111112232211111111111111111111111111111111...
   ...111111111111111111111111111111222111111111111111111111111111111111...
   ...111111111111111111111111111111121111111111111111111111111111111111...
   ...111111111111111111111111111111222111111111111111111111111111111111...
   ...111111111111111111111111111112232211111111111111111111111111111111...
   ...111111111111111111111111111122343221111111111111111111111111111111...
   ...111111111111111111111111111223454322111111111111111111111111111111...
   ...111111111111111111111111112234565432211111111111111111111111111111...
   ...111111111111111111111111122345676543221111111111111111111111111111...
   ...111111111111111111111111223456787654322111111111111111111111111111...
   ........................................................................




  12)Smarandache car:
   ...000000000000000000000000000000000000000000000000000000000000000000...
   ...000000000000000000111111111111111111111111100000000000000000000000...
   ...000000000000000001111111111111111111111111110000000000000000000000...
   ...000000000000000011000000000000000000000000011000000000000000000000...
   ...000000000000000110000000000000000000000000001100000000000000000000...
   ...000000011111111100000000000000000000000000000111111111111110000000...
   ...000000111111111000000000000000000000000000000011111111111111000000...
   ...000000110000000000000000000000000000000000000000000000000011200000...
   ...000000110000000000000000000000000000000000000000000000000011000000...
   ...000000110000044400000000000000000000000000000000000044400011000000...
   ...000000111111444441111111111111111111111111111111111444441111200000...
   ...000000011114444444111111111111111111111111111111114444444110000000...
   ...000000000000444440000000000000000000000000000000000444440000000000...
   ...000000000000044400000000000000000000000000000000000044400000000000...
   ...000000000000000000000000000000000000000000000000000000000000000000...
   ........................................................................





  13)Smarandache finite lattice:
   ...000000000000000000000000000000000000000000000000000000000000000000...
   ...077700000000000700000007777777700777777770077007777777700777777770...
   ...077700000000007770000007777777700777777770077007777777700777777770...
   ...077700000000077077000000007700000000770000077007770000000770000000...
   ...077700000000770007700000007700000000770000077007770000000777770000...
   ...077700000007777777700000007700000000770000077007770000000770000000...
   ...077777700077000000077000007700000000770000077007777777700777777770...
   ...077777700770000000007700007700000000770000077007777777700777777770...
   ...000000000000000000000000000000000000000000000000000000000000000000...
   ........................................................................




  14)Smarandache infinite lattice:
   ...111111111111111111111111111111111111111111111111111111111111111111...
   ...177711111111111711111117777777711777777771177117777777711777777771...
   ...177711111111117771111117777777711777777771177117777777711777777771...
   ...177711111111177177111111117711111111771111177117771111111771111111...
   ...177711111111771117711111117711111111771111177117771111111777771111...
   ...177711111117777777711111117711111111771111177117771111111771111111...
   ...177777711177111111177111117711111111771111177117777777711777777771...
   ...177777711771111111117711117711111111771111177117777777711777777771...
   ...111111111111111111111111111111111111111111111111111111111111111111...
   ........................................................................



   Remark:  of course, it's interesting to "design" a large variety of 
     numerical <object sequences> in the same way.
     Their numbers may be finite if the picture's background is zeroed, or 
     infinite if the picture's background is not zeroed -- as for the previous
     examples.
     


  15)Smarandache simple numbers:
    2,3,4,5,6,7,8,9,10,11,13,14,15,17,19,21,22,23,25,26,27,29,31,33,34,35,37,38,
    39,41,43,45,46,47,49,51,53,55,57,58,61,62,65,67,69,71,73,74,77,78,79,82,83,
    85,86,87,89,91,93,94,95,97,101,103,...
    (A number n is called <simple number> if the product of its proper divisors
     is less than or equal to n.)
    Generally speaking, n has the form:  
        n = p, or p^2, or p^3, or pq,  where p and q are distinct primes.


    References:
      Florentin Smarandache, "Only Problems, not Solutions!", Xiquan 
        Publishing House, Phoenix-Chicago, 1990, 1991, 1993; 
        ISBN: 1-879585-00-6.
        (reviewed in <Zentralblatt fur Mathematik> by P. Kiss: 11002, 
         pre744, 1992;
         and <The American Mathematical Monthly>, Aug.-Sept. 1991);
      Student Conference, University of Craiova, Department of Mathematics,
        April 1979,  "Some problems in number theory" by Florentin Smarandache.


  16)Smarandache digital sum:
     0,1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,10,2,3,4,5,6,7,8,9,10,11,
     |                 | |                  | |                   |
      -----------------   ------------------   -------------------

     3,4,5,6,7,8,9,10,11,12,4,5,6,7,8,9,10,11,12,13,5,6,7,8,9,10,11,12,13,14,...
     |                    | |                     | |                      |
      --------------------   ---------------------   ----------------------

     (d (n) is the sum of digits.)
       s


  17)Smarandache digital products:
     0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,2,4,6,8,19,12,14,16,18,
     |                 | |                 | |                      |
      -----------------   -----------------   ----------------------

     0,3,6,9,12,15,18,21,24,27,0,4,8,12,16,20,24,28,32,36,0,5,10,15,20,25,...
     |                       | |                        | |
      -----------------------   ------------------------   -------------- ...

     (d (n) is the product of digits.)
       p



  18)Smarandache code puzzle:
     151405,202315,2008180505,06152118,06092205,190924,1905220514,0509070820,
     14091405,200514,051205220514,...
    Using the following Smarandache letter-to-number code:
    A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z
   01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
   then c (n) = the numerical Smarandache code for the spelling of n in English
         p
   language;  for exemple: 1 = ONE = 151405, etc.


     
  19))Smarandache pierced chain:
     101,1010101,10101010101,101010101010101,1010101010101010101, 
     10101010101010101010101,101010101010101010101010101,...
     (c(n) = 101 * 1 0001 0001 ... 0001 , for n >= 1.)
                     |  | |  | ... |  |    
                     ---- ----     ----
                       1    2       n-1
      Smarandache asked how many c(n)/101 are primes ?


     References:
      Florentin Smarandache, "Only Problems, not Solutions!", Xiquan 
        Publishing House, Phoenix-Chicago, 1990, 1991, 1993; 
        ISBN: 1-879585-00-6.
        (reviewed in <Zentralblatt fur Mathematik> by P. Kiss: 11002, 
         pre744, 1992;
         and <The American Mathematical Monthly>, Aug.-Sept. 1991);
      Arizona State University, Hayden Library, "The Florentin Smarandache
        papers" special collection, Tempe, AZ 85287-1006, USA, phone: 
        (602)965-6515 (Carol Moore librarian), email: ICCLM@ASUACAD.BITNET . 
      Student Conference, University of Craiova, Department of Mathematics,
        April 1979,  "Some problems in number theory" by Florentin Smarandache.




  20)Smarandache divisor products:
     1,2,3,8,5,36,7,64,27,100,11,1728,13,196,225,1024,17,5832,19,8000,441,484,
     23,331776,125,676,729,21952,29,810000,31,32768,1089,1156,1225,10077696,37,
     1444,1521,2560000,41,...
     (P (n) is the product of all positive divisors of n.)
       d


  21)Smarandache proper divisor products:
     1,1,1,2,1,6,1,8,3,10,1,144,1,14,15,64,1,324,1,400,21,22,1,13824,5,26,27,
     784,1,27000,1,1024,33,34,35,279936,1,38,39,64000,1,...
     (p (n) is the product of all positive divisors of n but n.)
       d


  22)Smarandache square complements:
      1,2,3,1,5,6,7,2,1,10,11,3,14,15,1,17,2,19,5,21,22,23,6,1,26,3,7,29,30,31,
      2,33,34,35,1,37,38,39,10,41,42,43,11,5,46,47,3,1,2,51,13,53,6,55,14,57,58,
      59,15,61,62,7,1,65,66,67,17,69,70,71,2,...
     Definition:  
       for each integer n to find the smallest integer k such that 
       nk is a perfect square.. 
     (All these numbers are square free.)
      

  23)Smarandache cubic complements:
      1,4,9,2,25,36,49,1,3,100,121,18,169,196,225,4,289,12,361,50,441,484,529,
      9,5,676,1,841,900,961,2,1089,1156,1225,6,1369,1444,1521,25,1681,1764,1849,
      242,75,2116,2209,36,7,20,... 
     Definition:  
       for each integer n to find the smallest integer k such that 
       nk is a perfect cub. 
     (All these numbers are cube free.)
      

  24)Smarandache m-power complements:
     Definition:  
       for each integer n to find the smallest integer k such that 
       nk is a perfect m-power (m => 2). 
     (All these numbers are m-power free.)
     

    References:
      Florentin Smarandache, "Only Problems, not Solutions!", Xiquan 
        Publishing House, Phoenix-Chicago, 1990, 1991, 1993; 
        ISBN: 1-879585-00-6.
        (reviewed in <Zentralblatt fur Mathematik> by P. Kiss: 11002, 
         pre744, 1992;
         and <The American Mathematical Monthly>, Aug.-Sept. 1991);
      "The Florentin Smarandache papers" special collection, Arizona State 
         University, Hayden Library, Tempe, Box 871006, AZ 85287-1006, USA; 
         phone: (602) 965-6515 (Carol Moore & Marilyn Wurzburger: librarians), 
         email: ICCLM@ASUACAD.BITNET . 
      "The Encyclopedia of Integer Sequences", by N. J. A. Sloane and 
        S. Plouffe, Academic Press, San Diego, New York, Boston, London, 
        Sydney, Tokyo, Toronto, 1995;
        also online, email:  superseeker@research.att.com ( SUPERSEEKER by 
        N. J. A. Sloane, S. Plouffe, B. Salvy,  ATT Bell Labs, Murray Hill,
        NJ 07974, USA); 



  25)Smarandache cube free sieve:
     2,3,4,5,6,7,9,10,11,12,13,14,15,17,18,19,20,21,22,23,25,26,28,29,30,31,33,
     34,35,36,37,38,39,41,42,43,44,45,46,47,49,50,51,52,53,55,57,58,59,60,61,62,
     63,65,66,67,68,69,70,71,73,...
    Definition:  from the set of natural numbers (except 0 and 1):
       - take off all multiples of 2^3 (i.e. 8, 16, 24, 32, 40, ...)
       - take off all multiples of 3^3
       - take off all multiples of 5^3
       ... and so on (take off all multiples of all cubic primes).
    (One obtains all cube free numbers.)


  26)Smarandache m-power free sieve:
       Definition:  from the set of natural numbers (except 0 and 1)
         take off all multiples of 2^m, afterwards all multiples of 3^m, ...
         and so on (take off all multiples of all m-power primes, m >= 2).
       (One obtains all m-power free numbers.)


  27)Smarandache irrational root sieve:
     2,3,5,6,7,10,11,12,13,14,15,17,18,19,20,21,22,23,24,26,28,29,30,31,33,34, 
     35,37,38,39,40,41,42,43,44,45,46,47,48,50,51,52,53,54,55,56,57,58,59,60,61,
     62,63,65,66,67,68,69,70,71,72,73,...
    Definition:  from the set of natural numbers (except 0 and 1):
       - take off all powers of 2^k, k >= 2, (i.e. 4, 8, 16, 32, 64, ...)
       - take off all powers of 3^k, k >= 2;
       - take off all powers of 5^k, k >= 2;
       - take off all powers of 6^k, k >= 2;
       - take off all powers of 7^k, k >= 2;
       - take off all powers of 10^k, k >= 2;
       ... and so on (take off all k-powers, k >= 2, of all square free 
    numbers).
          He got all square free numbers by the following method (sieve):
          from the set of natural numbers (except 0 and 1):
          - take off all multiples of 2^2 (i.e. 4, 8, 12, 16, 20, ...)
          - take off all multiples of 3^2
          - take off all multiples of 5^2
          ... and so on (take off all multiples of all square primes);
          one obtains, therefore:
          2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,26,29,30,31,33,34,35,37,38,39,
          41,42,43,46,47,51,53,55,57,58,59,61,62,65,66,67,69,70,71,... ,
    which are used for Smarandache irrational root sieve.

    (One obtains all natural numbers those m-th roots, for any m >= 2, are 
     irrational.)



     References:
      Florentin Smarandache, "Only Problems, not Solutions!", Xiquan 
        Publishing House, Phoenix-Chicago, 1990, 1991, 1993; 
        ISBN: 1-879585-00-6.
        (reviewed in <Zentralblatt fur Mathematik> by P. Kiss: 11002, 
         pre744, 1992;
         and <The American Mathematical Monthly>, Aug.-Sept. 1991);
      Arizona State University, Hayden Library, "The Florentin Smarandache
        papers" special collection, Tempe, AZ 85287-1006, USA, phone: 
        (602)965-6515 (Carol Moore librarian), email: ICCLM@ASUACAD.BITNET . 
      Student Conference, University of Craiova, Department of Mathematics,
        April 1979,  "Some problems in number theory" by Florentin Smarandache.
      "The Encyclopedia of Integer Sequences", by N. J. A. Sloane and 
        S. Plouffe, Academic Press, San Diego, New York, Boston, London, 
        Sydney, Tokyo, Toronto, 1995;
        also online, email:  superseeker@research.att.com ( SUPERSEEKER by 
        N. J. A. Sloane, S. Plouffe, B. Salvy,  ATT Bell Labs, Murray Hill,
        NJ 07974, USA); 


  28)Smarandache odd sieve:
      7,13,19,23,25,31,33,37,43,47,49,53,55,61,63,67,73,75,79,83,85,91,93, 
      97,... 
      (All odd numbers that are not equal to the difference of two primes.)
       A sieve is used to get this sequence:
        - substract 2 from all prime numbers and obtain a temporary sequence;
        - choose all odd numbers that do not belong to the temporary one.


  29)Smarandache binary sieve:
     1,3,5,9,11,13,17,21,25,27,29,33,35,37,43,49,51,53,57,59,65,67,69,73,75,77,
     81,85,89,91,97,101,107,109,113,115,117,121,123,129,131,133,137,139,145,
     149,...
     (Starting to count on the natural numbers set at any step from 1:
        - delete every 2-nd numbers
        - delete, from the remaining ones, every 4-th numbers
        ... and so on:  delete, from the remaining ones, every (2^k)-th numbers,
        k = 1, 2, 3, ... .)
    
     Conjectures:  
       - there are an infinity of primes that belong to this sequence;
       - there are an infinity of numbers of this sequence which are not prime.



  30)Smarandache trinary sieve:
     1,2,4,5,7,8,10,11,14,16,17,19,20,22,23,25,28,29,31,32,34,35,37,38,41,43,46,
     47,49,50,52,55,56,58,59,61,62,64,65,68,70,71,73,74,76,77,79,82,83,85,86,88,
     91,92,95,97,98,100,101,103,104,106,109,110,112,113,115,116,118,119,122,124,
     125,127,128,130,131,133,137,139,142,143,145,146,149,...
     (Starting to count on the natural numbers set at any step from 1:
        - delete every 3-rd numbers
        - delete, from the remaining ones, every 9-th numbers
        ... and so on:  delete, from the remaining ones, every (3^k)-th numbers,
        k = 1, 2, 3, ... .)
    
     Conjectures:  
       - there are an infinity of primes that belong to this sequence;
       - there are an infinity of numbers of this sequence which are not prime.



  31)Smarandache n-ary sieve  (generalization, n >= 2):
     (Starting to count on the natural numbers set at any step from 1:
        - delete every n-th numbers
        - delete, from the remaining ones, every (n^2)-th numbers
        ... and so on:  delete, from the remaining ones, every (n^k)-th numbers,
        k = 1, 2, 3, ... .)
    

     Conjectures:  
       - there are an infinity of primes that belong to this sequence;
       - there are an infinity of numbers of this sequence which are not prime.

  32)Smarandache consecutive sieve:
     1,3,5,9,11,17,21,29,33,41,47,57,59,77,81,101,107,117,131,149,153,173,191,
     209,213,239,257,273,281,321,329,359,371,401,417,441,435,491,...
     (From the natural numbers set:
        - keep the first number,
          delete one number out of 2 from all remaining numbers;
        - keep the first remaining number,
          delete one number out of 3 from the next remaining numbers;
        - keep the first remaining number,
          delete one number out of 4 from the next remaining numbers;
        ... and so on, for step k (k >= 2):
        - keep the first remaining number,
          delete one number out of k from the next remaining numbers;
        ... .)

     This sequence is much less dense than the prime number sequence,
     and their ratio tends to p : n as n tends to infinity. 
                               n

     For this sequence we choosed to keep the first remaining number 
     at all steps,
     but in a more general case: 
     the kept number may be any among the remaining k-plet (even at random).
     


  33)Smarandache general-sequence sieve:
     Let u  > 1, for i = 1, 2, 3, ..., a stricly increasing positive integer  
          i
     sequence.  Then:
     From the natural numbers set:
       - keep one number among 1, 2, 3, ..., u - 1,
                                              1
         and delete every u -th numbers;
                           1
       - keep one number among the next u - 1 remaining numbers,
                                         2
         and delete every u -th numbers;
                           2
       ... and so on, for step k (k >= 1):
       - keep one number among the next u - 1 remaining numbers,
                                         k
         and delete every u -th numbers;
                           k
       ... .


     Problem:  study the relationship between sequence u , i = 1, 2, 3, ...,
                                                        i
     and the remaining sequence resulted from the Smarandache general 
     sieve.

     u , previously defined, is called Smarandache sieve generator.
      i


  34)Smarandache more general-sequence sieve:
     For i = 1, 2, 3, ..., let u  > 1, be a stricly increasing positive integer
                                i        
     sequence, and v  <  u  another positive integer sequence.  Then:
                    i     i
     From the natural numbers set:
       - keep the v -th number among 1, 2, 3, ..., u - 1,
                   1                                1
         and delete every u -th numbers;
                           1
       - keep the v -th number among the next u - 1 remaining numbers,
                   2                           2
         and delete every u -th numbers;
                           2
       ... and so on, for step k (k >= 1):
       - keep the v -th number among the next u - 1 remaining numbers,
                   k                           k
         and delete every u -th numbers;
                           k
       ... .


     Problem:  study the relationship between sequences u , v , i = 1, 2, 3, 
                                                         i   i
     ..., and the remaining sequence resulted from the Smarandache more general 
     sieve.

     u  and v  previously defined, are called Smarandache sieve generators.
      i      i


  35)Smarandache digital sequences:
      (This a particular case of Smarandache sequences of sequences.)
      General definition:
      in any numeration base B, for any given infinite integer or rational 
      sequence S , S , S , ...,  and any digit D from 0 to B-1, 
                1   2   3
      it's built up a new integer sequence witch
        associates to S the number of digits D of S  in base B, 
                       1                           1
        to S  the number of digits D of S  in base B,  and so on...
            2                            2

      For exemple, considering the prime number sequence in base 10, 
      then the number of digits 1 (for exemple) of each prime number 
      following their order is: 0,0,0,0,2,1,1,1,0,0,1,0,...
      (Smarandache digit-1 prime sequence).

      Second exemple if we consider the factorial sequence n! in base 10,
      then the number of digits 0 of each factorial number 
      following their order is: 0,0,0,0,0,1,1,2,2,1,3,...
      (Smarandache digit-0 factorial sequence). 

      Third exemple if we consider the sequence n^n in base 10, n=1,2,...,
      then the number of digits 5 of each term 1^1, 2^2, 3^3, ...,
      following their order is: 0,0,0,1,1,1,1,0,0,0,...
      (Smarandache digit-5 n^n sequence).
                                                

     References: 
      E. Grosswald, University of Pennsylvania, Philadelphia, Letter to F.
        Smarandache, August 3, 1985;
      R. K. Guy, University of Calgary, Alberta, Canada, Letter to F. 
        Smarandache, November 15, 1985;
      Florentin Smarandache, "Only Problems, not Solutions!", Xiquan 
        Publishing House, Phoenix-Chicago, 1990, 1991, 1993; 
        ISBN: 1-879585-00-6.
        (reviewed in <Zentralblatt fur Mathematik> by P. Kiss: 11002, 
         pre744, 1992;
         and <The American Mathematical Monthly>, Aug.-Sept. 1991);
      Arizona State University, Hayden Library, "The Florentin Smarandache
        papers" special collection, Tempe, AZ 85287-1006, USA, phone: 
        (602)965-6515 (Carol Moore librarian), email: ICCLM@ASUACAD.BITNET . 



  35)Smarandache construction sequences:
      (This a particular case of Smarandache sequences of sequences.)
      General definition:
      in any numeration base B, for any given infinite integer or rational 
      sequence S , S , S , ...,  and any digits D , D , ..., D  (k < B),
                1   2   3                        1   2        k
      it's built up a new integer sequence such that
        each of its terms Q  < Q  < Q  < ... is formed by these digits 
                           1    2    3
        D , D , ..., D  only (all these digits are used), and matches a 
         1   2        k
        term S  of the previous sequence.
              i

      For exemple, considering in base 10 the prime number sequence,
      and the digits 1 and 7 (for exemple),
      we construct a written-only-with-these-digits (all these digits are used)
      prime number new sequence:  17,71,...
      (The Smarandache digit-1-7-only prime sequence).

      Second exemple, considering in base 10 the multiple of 3 sequence,
      and the digits 0 and 1,
      we construct a written-only-with-these-digits (all these digits are
      used) multiple of 3 new sequence:  1011,1101,1110,10011,10101,10110,
      11001,11010,11100,...
      (The Smarandache digit-0-1-only multiple of 3 sequence).


     References: 
      E. Grosswald, University of Pennsylvania, Philadelphia, Letter to F.
        Smarandache, August 3, 1985;
      R. K. Guy, University of Calgary, Alberta, Canada, Letter to F. 
        Smarandache, November 15, 1985;
      Florentin Smarandache, "Only Problems, not Solutions!", Xiquan 
        Publishing House, Phoenix-Chicago, 1990, 1991, 1993; 
        ISBN: 1-879585-00-6.
        (reviewed in <Zentralblatt fur Mathematik> by P. Kiss: 11002, 
         pre744, 1992;
         and <The American Mathematical Monthly>, Aug.-Sept. 1991);
      Arizona State University, Hayden Library, "The Florentin Smarandache
        papers" special collection, Tempe, AZ 85287-1006, USA, phone: 
        (602)965-6515 (Carol Moore librarian), email: ICCLM@ASUACAD.BITNET . 



  36)Smarandache general residual sequence:
      (x + C )...(x + C    ), m = 2, 3, 4, ...,
            1          F(m)
        where C , 1 <= i <= F(m), forms a reduced set of residues mod m,
               i
        x is an integer, and F is Euler's totient.
      The Smarandache General Residual Sequence is induced from the 
        The Smarandache Residual Function (see <Libertas Mathematica>):
        Let L : ZxZ --> Z be a function defined by
            L(x,m)=(x + C )...(x + C    ),
                         1          F(m)
        where C , 1 <= i <= F(m), forms a reduced set of residues mod m,
               i
        m >= 2, x is an integer, and F is Euler's totient.
      The Smarandache Residual Function is important because it generalizes
      the classical theorems by Wilson, Fermat, Euler, Wilson, Gauss, Lagrange,
      Leibnitz, Moser, and Sierpinski all together.
      For x=0 it's obtained the following sequence:
        L(m) = C  ... C    , where m = 2, 3, 4, ...
                1      F(m)
      (the product of all residues of a reduced set mod m):
      1,2,3,24,5,720,105,2240,189,3628800,385,479001600,19305,896896,2027025,
      20922789888000,85085,6402373705728000,8729721,47297536000,1249937325,...
      which is found in "The Enciclopedia of Integer Sequences", by N. J. A.
      Sloane, Academic Press, San Diego, New York, Boston, London, 
      Sydney, Tokyo, Toronto, 1995. 
      The Smarandache Residual Function extends it.

     References:
      Fl. Smarandache, "A numerical function in the congruence theory", in
        <Libertah Mathematica>, Texas State University, Arlington, 12,
        pp. 181-185, 1992;
        see <Mathematical Reviews> 93i:11005 (11A07), p.4727,
        and <Zentralblatt fur Mathematik>, Band 773(1993/23), 11004 (11A);
      Fl. Smarandache, "Collected Papers" (Vol. 1), Ed. Tempus, Bucharest, 
        1995;      
      Arizona State University, Hayden Library, "The Florentin Smarandache
        papers" special collection, Tempe, AZ 85287-1006, USA, phone: 
        (602)965-6515 (Carol Moore librarian), email: ICCLM@ASUACAD.BITNET . 
      Student Conference, University of Craiova, Department of Mathematics,
        April 1979,  "Some problems in number theory" by Florentin Smarandache.



  37)Smarandache (inferior) prime part:
      2,3,3,5,5,7,7,7,7,11,11,13,13,13,13,17,17,19,19,19,19,23,23,23,23,23,23,
      29,29,31,31,31,31,31,31,37,37,37,37,41,41,43,43,43,43,47,47,47,47,47,47,
      53,53,53,53,53,53,59, ...
      (For any positive real number n one defines p (n) as the largest prime 
                                                   p
       number less than or equal to n.)


  38)Smarandache (superior) prime part:
      2,2,2,3,5,5,7,7,11,11,11,11,13,13,17,17,17,17,19,19,23,23,23,23,29,29,29,
      29,29,29,31,31,37,37,37,37,37,37,41,41,41,41,43,43,47,47,47,47,53,53,53, 
      53,53,53,59,59,59,59,59,59,61,...
      (For any positive real number n one defines P (n) as the smallest prime 
                                                   p
       number greater than or equal to n.)


    References:
      Florentin Smarandache, "Only Problems, not Solutions!", Xiquan 
        Publishing House, Phoenix-Chicago, 1990, 1991, 1993; 
        ISBN: 1-879585-00-6.
        (reviewed in <Zentralblatt fur Mathematik> by P. Kiss: 11002, 
         pre744, 1992;
         and <The American Mathematical Monthly>, Aug.-Sept. 1991);
      "The Florentin Smarandache papers" special collection, Arizona State 
         University, Hayden Library, Tempe, Box 871006, AZ 85287-1006, USA; 
         phone: (602) 965-6515 (Carol Moore & Marilyn Wurzburger: librarians), 
         email: ICCLM@ASUACAD.BITNET . 
      "The Encyclopedia of Integer Sequences", by N. J. A. Sloane and 
        S. Plouffe, Academic Press, San Diego, New York, Boston, London, 
        Sydney, Tokyo, Toronto, 1995;
        also online, email:  superseeker@research.att.com ( SUPERSEEKER by 
        N. J. A. Sloane, S. Plouffe, B. Salvy,  ATT Bell Labs, Murray Hill,
        NJ 07974, USA); 



  39)Smarandache (inferior) square part:
     0,1,1,1,4,4,4,4,4,9,9,9,9,9,9,9,16,16,16,16,16,16,16,16,16,25,25,25,25,25,
     25,25,25,25,25,25,36,36,36,36,36,36,36,36,36,36,36,36,36,49,49,49,49,49,
     49,49,49,49,49,49,49,49,49,49,64,64,...
     (The largest square less than or equal to n.)


  40)Smarandache (superior) square part:
     0,1,4,4,4,9,9,9,9,9,16,16,16,16,16,16,16,25,25,25,25,25,25,25,25,25,36,36,
     36,36,36,36,36,36,36,36,36,49,49,49,49,49,49,49,49,49,49,49,49,49,64,64,64,
     64,64,64,64,64,64,64,64,64,64,64,64,81,81,...
     (The smallest square greater than or equal to n.)


  41)Smarandache (inferior) cube part:
     0,1,1,1,1,1,1,1,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,27,27,27,27,27,27,
     27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,
     27,27,27,27,27,27,27,64,64,64,...
     (The largest cube less than or equal to n.)


  42)Smarandache (superior) cube part:
     0,1,8,8,8,8,8,8,8,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,
     27,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,64,
     64,64,64,64,64,64,64,64,64,64,64,64,64,64,125,125,125,..
     (The smalest cube greater than or equal to n.)


  43)Smarandache (inferior) factorial part:
     1,2,2,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,24,24,24,24,24,24,24,24,24,
     24,24,24,24,24,24,24,24,24,...
     (F (n) is the largest factorial less than or equal to n.)
       p


  44)Smarandache (superior) factorial part:
     1,2,6,6,6,6,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,120,120,
     120,120,120,120,120,120,120,120,120,...
     (f (n) is the smallest factorial greater than or equal to n.)
       p


  45)Smarandache double factorial complements:
      1,1,1,2,3,8,15,1,105,192,945,4,10395,46080,1,3,2027025,2560,34459425,192,
      5,3715891200,13749310575,2,81081,1961990553600,35,23040,213458046676875,
      128,6190283353629375,12,...
      (For each n to find the smallest k such that nk is a double factorial,
       i.e. nk = either 1*3*5*7*9*...*n if n is odd, 
                 either 2*4*6*8*...*n if n is even.)


  46)Smarandache prime additive complements:
      1,0,0,1,0,1,0,3,2,1,0,1,0,3,2,1,0,1,0,3,2,1,0,5,4,3,2,1,0,1,0,5,4,3,2,1,0,
      3,2,1,0,1,0,3,2,1,0,5,4,3,2,1,0,...
      (For each n to find the smallest k such that n+k is prime.)
     Remark:  Smarandache asked if it's possible to get as large as we want
       but finite decreasing k, k-1, k-2, ..., 2, 1, 0 (odd k) sequence
       included in the previous sequence -- i.e. for any even integer are 
       there two primes those difference is equal to it?  He conjectured the 
       answer is negative.


    References:
      Florentin Smarandache, "Only Problems, not Solutions!", Xiquan 
        Publishing House, Phoenix-Chicago, 1990, 1991, 1993; 
        ISBN: 1-879585-00-6.
        (reviewed in <Zentralblatt fur Mathematik> by P. Kiss: 11002, 
         pre744, 1992;
         and <The American Mathematical Monthly>, Aug.-Sept. 1991);
      "The Florentin Smarandache papers" special collection, Arizona State 
         University, Hayden Library, Tempe, Box 871006, AZ 85287-1006, USA; 
         phone: (602) 965-6515 (Carol Moore & Marilyn Wurzburger: librarians), 
         email: ICCLM@ASUACAD.BITNET . 



  47)Smarandache prime base:
      0,1,10,100,101,1000,1001,10000,10001,10010,10100,100000,100001,1000000, 
      1000001,1000010,1000100,10000000,10000001,100000000,100000001,100000010,
      100000100,1000000000,1000000001,1000000010,1000000100,1000000101,...
      (Each number n written in the Smarandache prime base.)

    (Smarandache defined over the set of natural numbers the following infinite
     base:  p  = 1, and for k >= 1  p  is the k-th prime number.)
             0                       k
     He proved that every positive integer A may be uniquely written in
     the Smarandache prime base as:
                                n
         ___________      def  ---
    A = (a  ... a a )     ===  \  a p  , with all a = 0 or 1, (of course a = 1),
          n      1 0 (SP)      /   i i             i                      n
                               ---
                               i=0
    in the following way:
      - if  p  <= A  < p    then A  = p  + r ;
             n          n+1            n    1
      - if  p  <= r  < p    then r  = p  + r , m < n;
             m     1    m+1       1    m    2
      and so on untill one obtains a rest r  = 0.
                                           j
    Therefore, any number may be written as a sum of prime numbers + e,
    where e = 0 or 1.

    If we note by p(A) the Smarandache superior part of A (i.e. the largest
    prime less than or equal to A), then
    A is written in the Smarandache prime base as:

        A = p(A) + p(A-p(A)) + p(A-p(A)-p(A-p(A))) + ... .

    This base is important for partitions with primes. 
                               


  48)Smarandache square base:
     0,1,2,3,10,11,12,13,20,100,101,102,103,110,111,112,1000,1001,1002,1003,
     1010,1011,1012,1013,1020,10000,10001,10002,10003,10010,10011,10012,10013,
     10020,10100,10101,100000,100001,100002,100003,100010,100011,100012,100013,
     100020,100100,100101,100102,100103,100110,100111,100112,101000,101001,
     101002,101003,101010,101011,101012,101013,101020,101100,101101,101102,
     1000000,...
   (Each number n written in the Smarandache square base.)

   (Smarandache defined over the set of natural numbers the following infinite
    base:  for k >= 0  s  = k^2.)
                        k

    He proved that every positive integer A may be uniquely written in
    the Smarandache square base as:
                                 n
          ___________      def  ---
     A = (a  ... a a )     ===  \  a s  , with a = 0 or 1 for i >= 2, 
           n      1 0 (S2)      /   i i         i
                                ---
                                i=0
   
     0 <= a  <= 3,  0 <= a  <= 2, and of course a  = 1,
           0              1                      n
    in the following way:
      - if  s  <= A  < s    then A  = s  + r ;
             n          n+1            n    1
      - if  s  <= r  < p    then r  = s  + r , m < n;
             m     1    m+1       1    m    2
      and so on untill one obtains a rest r  = 0.
                                           j

    Therefore, any number may be written as a sum of squares (1 not counted
    as a square -- being obvious) + e, where e = 0, 1, or 3.

    If we note by s(A) the Smarandache superior square part of A (i.e. the 
    largest square less than or equal to A), then A is written in the 
    Smarandache square base as:

        A = s(A) + s(A-s(A)) + s(A-s(A)-s(A-s(A))) + ... .

    This base is important for partitions with squares. 
                               

 
  49)Smarandache m-power base  (generalization):
      (Each number n written in the Smarandache m-power base, 
       where m is an integer >= 2.)

     (Smarandache defined over the set of natural numbers the following infinite
      m-power base:  for k >= 0  t  = k^m.)
                                k

      He proved that every positive integer A may be uniquely written in
      the Smarandache m-power base as:
                                 n
          ___________      def  ---
     A = (a  ... a a )     ===  \  a t  , with a = 0 or 1 for i >= m, 
           n      1 0 (SM)      /   i i         i
                                ---
                                i=0
                 --                       --
      0 <= a  <= | ((i+2)^m - 1) / (i+1)^m | (integer part)
            i    --                       --   
     for i = 0, 1, ..., m-1, a  = 0 or 1 for i >= m, and of course a  = 1,
                              i                                     n
    in the following way:
      - if  t  <= A  < t    then A  = t  + r ;
             n          n+1            n    1
      - if  t  <= r  < t    then r  = t  + r , m < n;
             m     1    m+1       1    m    2
      and so on untill one obtains a rest r  = 0.
                                           j

    Therefore, any number may be written as a sum of m-powers (1 not counted
    as an m-power -- being obvious) + e, where e = 0, 1, 2, ..., or 2^m-1.

    If we note by t(A) the Smarandache superior m-power part of A (i.e. the 
    largest m-power less than or equal to A), then A is written in the 
    Smarandache m-power base as:

        A = t(A) + t(A-t(A)) + t(A-t(A)-t(A-t(A))) + ...

    This base is important for partitions with m-powers. 
                               

 
  50)Smarandache factorial base:
    0,1,10,11,20,21,100,101,110,111,120,121,200,201,210,211,220,221,300,301,310,
    311,320,321,1000,1001,1010,1011,1020,1021,1100,1101,1110,1111,1120,1121,
    1200,...
   (Each number n written in the Smarandache factorial base.)

   (Smarandache defined over the set of natural numbers the following infinite
    base:  for k >= 1  f  = k!)
                        k

    He proved that every positive integer A may be uniquely written in
    the Smarandache square base as:
                                 n
          ___________      def  ---
     A = (a  ... a a )     ===  \  a f  , with all a = 0, 1, ..., i for i >= 1. 
           n      2 1 (F)       /   i i             i
                                ---
                                i=1
   
    in the following way:
      - if  f  <= A  < f    then A  = f  + r ;
             n          n+1            n    1
      - if  f  <= r  < f    then r  = f  + r , m < n;
             m     1    m+1       1    m    2
      and so on untill one obtains a rest r  = 0.
                                           j

    What's very interesting:  a  = 0 or 1;  a = 0, 1, or 2;  a = 0, 1, 2, or 3,
                               1             2                3
    and so on...

    If we note by f(A) the Smarandache superior factorial part of A (i.e. the 
    largest factorial less than or equal to A), then A is written in the 
    Smarandache factorial base as:

        A = f(A) + f(A-f(A)) + f(A-f(A)-f(A-f(A))) + ... .

    Rules of addition and subtraction in Smarandache factorial base:
    foreach digit a  we add and substract in base i+1, for i >= 1.
                   i
    For example, an addition:
                               base  5 4 3 2
                               ---------------
                                       2 1 0 +
                                       2 2 1
                                   -----------
                                     1 1 0 1
    because: 0+1= 1   (in base 2);
             1+2=10   (in base 3), therefore we write 0 and keep 1;
             2+2+1=11 (in base 4).

    Now a subtraction:
                               base  5 4 3 2
                               ---------------
                                     1 0 0 1 -
                                       3 2 0
                                     ---------
                                     = = 1 1
    because: 1-0=1 (in base 2);
             0-2=? it's not possible (in base 3), 
                   go to the next left unit, which is 0 again (in base 4),
                   go again to the next left unit, which is 1 (in base 5),
                   therefore 1001 --> 0401 --> 0331
                   and then 0331-320=11.

    Find some rules for multiplication and division.


    In a general case:
      if we want to design a base such that any number
                                 n
          ___________      def  ---
     A = (a  ... a a )     ===  \  a b  , with all a = 0, 1, ..., t  for 
           n      2 1 (B)       /   i i             i              i
                                ---
                                i=1
     i >= 1, where all t >= 1, then:
                        i 
     this base should be
     b = 1,  b    = (t +1) * b  for i >= 1.
      1       i+1     i       i


 
  51)Smarandache generalized base:
     (Each number n written in the Smarandache generalized base.) 

     (Smarandache defined over the set of natural numbers the following infinite
      generalized base:  1 = g  <  g  < ... < g  < ... .)
                              0     1          k
                                
      He proved that every positive integer A may be uniquely written in
      the Smarandache generalized base as:
                                 n
          ___________      def  ---                       --                --
     A = (a  ... a a )     ===  \  a g  , with 0 <= a  <= | (g   - 1) /  g   |
           n      1 0 (SG)      /   i i              i    --  i+1         i --
                                ---
                                i=0
    (integer part) for i = 0, 1, ..., n,  and of course a  >= 1,
                                                         n    
    in the following way:
      - if  g  <= A  < g    then A  = g  + r ;
             n          n+1            n    1
      - if  g  <= r  < g    then r  = g  + r ,  m < n;
             m     1    m+1       1    m    2
      and so on untill one obtains a rest r  = 0.
                                           j
    If we note by g(A) the Smarandache superior generalized part of A (i.e. the 
    largest g  less than or equal to A), then A is written in the 
             i
    Smarandache m-power base as:

        A = g(A) + g(A-g(A)) + g(A-g(A)-g(A-g(A))) + ...

    This base is important for partitions:  the generalized base may be any
    infinite integer set (primes, squares, cubes, any m-powers, Fibonacci/Lucas 
    numbers, Bernoully numbers, Smarandache numbers, etc.) those partitions are
    studied.
      A particular case is when the base verifies: 2g  >= g    for any i,
                                                     i     i+1 
    and g  = 1, because all coefficients of a written number in this base 
         0
    will be 0 or 1.

                                                            i-1
       Remark:  another particular case: if one takes g  = p   , i = 1, 2, 3,
                                                       i
      ..., p an integer >= 2, one gets the representation of a number in the 
      numerical base p {p may be 10 (decimal), 2 (binar), 16 (hexadecimal), 
      etc.}.


  52)Smarandache numbers:
      1,2,3,4,5,3,7,4,6,5,11,4,13,7,5,6,17,6,19,5,7,11,23,4,10,13,9,7,29,
      5,31,8,11,17,7,6,37,19,13,5,41,7,43,11,5,23,47,6,14,10,17,13,53,9,11,
      7,19,29,59,5,61,31,7,8,13,...      
      (S(n) is the smallest integer such that S(n)! is divisible by n.)
       Remark:  S(n) are the values of Smarandache function.


  53)Smarandache quotients:
      1,1,2,6,24,1,720,3,80,12,3628800,2,479001600,360,8,45,20922789888000,
      40,6402373705728000,6,240,1814400,1124000727777607680000,1,145152,
      239500800,13440,180,304888344611713860501504000000,...
      (For each n to find the smallest k such that nk is a factorial number.)


    References:
      "The Florentin Smarandache papers" special collection, Arizona State 
         University, Hayden Library, Tempe, Box 871006, AZ 85287-1006, USA; 
         phone: (602) 965-6515 (Carol Moore & Marilyn Wurzburger: librarians), 
         email: ICCLM@ASUACAD.BITNET . 
      "The Encyclopedia of Integer Sequences", by N. J. A. Sloane and 
        S. Plouffe, Academic Press, San Diego, New York, Boston, London, 
        Sydney, Tokyo, Toronto, 1995;
        also online, email:  superseeker@research.att.com ( SUPERSEEKER by 
        N. J. A. Sloane, S. Plouffe, B. Salvy,  ATT Bell Labs, Murray Hill,
        NJ 07974, USA); 



  54)Smarandache double factorial numbers:
      1,2,3,4,5,6,7,4,9,10,11,6,13,14,5,6,17,12,19,10,7,22,23,6,15,26,9,14,29, 
      10,31,8,11,34,7,12,37,38,13,10,41,14,43,22,9,46,47,6,21,10,...
      (d (n) is the smallest integer such that d (n)!! is a multiple of n.)
        f                                       f


    References:
      Florentin Smarandache, "Only Problems, not Solutions!", Xiquan 
        Publishing House, Phoenix-Chicago, 1990, 1991, 1993; 
        ISBN: 1-879585-00-6.
        (reviewed in <Zentralblatt fur Mathematik> by P. Kiss: 11002, 
         pre744, 1992;
         and <The American Mathematical Monthly>, Aug.-Sept. 1991);
      "The Florentin Smarandache papers" special collection, Arizona State 
         University, Hayden Library, Tempe, Box 871006, AZ 85287-1006, USA; 
         phone: (602) 965-6515 (Carol Moore & Marilyn Wurzburger: librarians), 
         email: ICCLM@ASUACAD.BITNET . 



  55)Smarandache primitive numbers (of power 2):
     2,4,4,6,8,8,8,10,12,12,14,16,16,16,16,18,20,20,22,24,24,24,26,28,28,30,32, 
     32,32,32,32,34,36,36,38,40,40,40,42,44,44,46,48,48,48,48,50,52,52,54,56,56,
     56,58,60,60,62,64,64,64,64,64,64,66,...
     (S (n) is the smallest integer such that S (n)! is divisible by 2^n.)
       2                                       2

     Curious property:  this is the sequence of even numbers, each number being
     repeated as many times as its exponent (of power 2) is.

     This is one of Smarandache irreductible functions, noted S (k), which helps 
                                                               2
     to calculate the Smarandache function (called also Smarandache numbers 
     in "The Encyclopedia of Integer Sequences", by N. J. A. Sloane and S. 
     Plouffe, Academic Press, San Diego, New York, Boston, London, Sydney, 
     Tokyo, Toronto, 1995).


  56)Smarandache primitive numbers (of power 3):
     3,6,9,9,12,15,18,18,21,24,27,27,27,30,33,36,36,39,42,45,45,48,51,54,54,54, 
     57,60,63,63,66,69,72,72,75,78,81,81,81,81,84,87,90,90,93,96,99,99,102,105,
     108,108,108,111,...
     (S (n) is the smallest integer such that S (n)! is divisible by 3^n.)
       3                                       3
         
     Curious property:  this is the sequence of multiples of 3, each 
     number being repeated as many times as its exponent (of power 3) is.

     This is one of Smarandache irreductible functions, noted S (k), which helps
                                                               3
     to calculate the Smarandache function (called also Smarandache numbers
     in "The Encyclopedia of Integer Sequences", by N. J. A. Sloane and S.
     Plouffe, Academic Press, San Diego, New York, Boston, London, Sydney, 
     Tokyo, Toronto, 1995).



  57)Smarandache primitive numbers (of power p, p prime)  {generalization}:
      (S (n) is the smallest integer such that S (n)! is divisible by p^n.)
        p                                       p

      Curious property:  this is the sequence of multiples of p, each
      number being repeated as many times as its exponent (of power p) is.

      These are the Smarandache irreductible functions, noted S (k), for any 
                                                               p
      prime number p, which helps to calculate the Smarandache function (called 
      also Smarandache numbers in "The Encyclopedia of Integer Sequences", by N.
      J. A. Sloane and S. Plouffe, Academic Press, San Diego, New York, Boston,
      London, Sydney, Tokyo, Toronto, 1995).


  58)Smarandache square residues:
    1,2,3,2,5,6,7,2,3,10,11,6,13,14,15,2,17,6,19,10,21,22,23,6,5,26,3,14,29,30,
    31,2,33,34,35,6,37,38,39,10,41,42,43,22,15,46,47,6,7,10,51,26,53,6,14,57,58,
    59,30,61,62,21,...
    (s (n) is the largest square free number which divides n.)
      r

     Or, s (n) is the number n released of its squares:
          r
     if n = (p ^ a ) * ... * (p ^ a ), with all p  primes and all a  >= 1,
              1   1            r   r             i                 i
     then s (n) = p * ... * p .
           r       1         r

     Remark:  at least the (2^2)*k-th numbers (k = 1, 2, 3, ...) are released
     of their squares;
     and more general:  all (p^2)*k-th numbers (for all p prime, and k = 1, 2, 
     3, ...) are released of their squares.


  59)Smarandache cubical residues:
    1,2,3,4,5,6,7,4,9,10,11,12,13,14,15,4,17,18,19,20,21,22,23,12,25,26,9,28, 
    29,30,31,4,33,34,35,36,37,38,39,20,41,42,43,44,45,46,47,12,49,50,51,52,53, 
    18,55,28,...
    (c (n) is the largest cube free number which divides n.)
      r

     Or, c (n) is the number n released of its cubicals:
          r
     if n = (p ^ a ) * ... * (p ^ a ), with all p  primes and all a  >= 1,
              1   1            r   r             i                 i
     then c (n) = (p ^ b ) * ... * (p ^ b ), with all b  = min {2, a }.  
           r        1   1            r   r             i            i

     Remark:  at least the (2^3)*k-th numbers (k = 1, 2, 3, ...) are released
     of their cubicals;
     and more general:  all (p^3)*k-th numers (for all p prime, and k = 1, 2, 
     3, ...) are released of their cubicals.



  60)Smarandache m-power residues   (generalization):
      (m (n) is the largest m-power free number which divides n.)
        r

     Or, m (n) is the number n released of its m-powers:
          r
     if n = (p ^ a ) * ... * (p ^ a ), with all p  primes and all a  >= 1,
              1   1            r   r             i                 i
     then m (n) = (p ^ b ) * ... * (p ^ b ), with all b  = min { m-1, a }.  
           r        1   1            r   r             i               i

     Remark:  at least the (2^m)*k-th numbers (k = 1, 2, 3, ...) are released
     of their m-powers;
     and more general:  all (p^m)*k-th numers (for all p prime, and k = 1, 2, 
     3, ...) are released of their m-powers.



  61)Smarandache exponents (of power 2):
     0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,2,0,1,0,2,0,1,0,5,0,1,0,2,0, 
     1,0,3,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,6,0,1,...
     (e (n) is the largest exponent (of power 2) which divides n.)
       2

     Or, e (n) = k  if 2^k divides n but 2^(k+1) does not.    
          2


  62)Smarandache exponents (of power 3):
    0,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,0,3,0,0,1,0,0,1,0,0,2,0,
    0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,0,2,0,0,1,0,...    
    (e (n) is the largest exponent (of power 3) which divides n.)
      3

    Or, e (n) = k  if 3^k divides n but 3^(k+1) does not.    
         3


  63)Smarandache exponents (of power p)  {generalization}:
      (e (n) is the largest exponent (of power p) which divides n,
        p
       where p is an integer >= 2.)

       Or, e (n) = k  if p^k divides n but p^(k+1) does not.    
            p


    References:
      Florentin Smarandache, "Only Problems, not Solutions!", Xiquan 
        Publishing House, Phoenix-Chicago, 1990, 1991, 1993; 
        ISBN: 1-879585-00-6.
        (reviewed in <Zentralblatt fur Mathematik> by P. Kiss: 11002, 
         pre744, 1992;
         and <The American Mathematical Monthly>, Aug.-Sept. 1991);
      Arizona State University, Hayden Library, "The Florentin Smarandache
        papers" special collection, Tempe, AZ 85287-1006, USA, phone: 
        (602)965-6515 (Carol Moore librarian), email: ICCLM@ASUACAD.BITNET . 



  64)Smarandache pseudo-primes of first kind:
     2,3,5,7,11,13,14,16,17,19,20,23,29,30,31,32,34,35,37,38,41,43,47,50,53,59,
     61,67,70,71,73,74,76,79,83,89,91,92,95,97,98,101,103,104,106,107,109,110,
     112,113,115,118,119,121,124,125,127,128,130,131,133,134,136,137,139,140,
     142,143,145,146, ...
     (A number is a Smarandache pseudo-prime of first kind if some permutation
      of the digits is a prime number, including the identity permutation.)    

     (Of course, all primes are Smarandache pseudo-primes of first kind, 
      but not the reverse!)


  65)Smarandache pseudo-primes of second kind:
      14,16,20,30,32,34,35,38,50,70,74,76,91,92,95,98,104,106,110,112,115,118,
      119,121,124,125,128,130,133,134,136,140,142,143,145,146, ...
      (A composite number is a Smarandache pseudo-prime of second kind if some 
       permutation of the digits is a prime number.)    


  66)Smarandache pseudo-primes of third kind:
     11,13,14,16,17,20,30,31,32,34,35,37,38,50,70,71,73,74,76,79,91,92,95,97,98,
     101,103,104,106,107,109,110,112,113,115,118,119,121,124,125,127,128,130,
     131,133,134,136,137,139,140,142,143,145,146, ...
     (A number is a Smarandache pseudo-prime of third kind if some nontrivial
      permutation of the digits is a prime number.)    

     Question:  How many Smarandache pseudo-primes of third kind are prime 
       numbers?  (he conjectured: an infinity).
     (There are primes which are not Smarandache pseudo-primes of third kind, 
      and the reverse:
      there are Smarandache pseudo-primes of third kind which are not primes.)


  67)Smarandache almost primes of first kind:
     a  >= 2, and for n >= 1  a    is the smallest number that is not divisible 
      1                        n+1
     by any of the previous terms (of the sequence) a , a , ..., a  .
                                                     1   2        n

     Example for a = 10:
                  1
       10,11,12,13,14,15,16,17,18,19,21,23,25,27,29,31,35,37,41,43,47,49,53,57,
       61,67,71,73,...

       If one starts by a  = 2, it obtains the complete prime sequence and only 
                         1
     it.
                                                                            2
       If one starts by a  > 2, it obtains after a rank r, where a  = p( a )
                         1                                        r       1
     with p(x) the strictly superior prime part of x, i.e. the largest prime 
     strictly less than x, the prime sequence:
     - between a  and a , the sequence contains all prime numbers of this 
                1      r
       interval and some composite numbers;
     - from a    and up, the sequence contains all prime numbers greater than 
             r+1
       a  and no composite numbers.
        r

  68)Smarandache almost primes of second kind:
     a  >= 2, and for n >= 1  a    is the smallest number that is coprime 
      1                        n+1
     with all of the previous terms (of the sequence) a , a , ..., a  .
                                                       1   2        n

     This second kind sequence merges faster to the prime numbers than
     the first kind sequence.


     Example for a = 10:
                  1
       10,11,13,17,19,21,23,29,31,37,41,43,47,53,57,61,67,71,73,...

       If one starts by a  = 2, it obtains the complete prime sequence and only 
                         1
     it.
       If one starts by a  > 2, it obtains after a rank r, where a  = p p
                         1                                        r    i j
     with p  and p  prime numbers strictly less than and not dividing a ,
           i      j                                                    1
     the prime sequence:
     - between a  and a , the sequence contains all prime numbers of this 
                1      r
       interval and some composite numbers;
     - from a    and up, the sequence contains all prime numbers greater than 
             r+1
       a  and no composite numbers.
        r


  69)Smarandache pseudo-squares of first kind:
    1,4,9,10,16,18,25,36,40,46,49,52,61,63,64,81,90,94,100,106,108,112,121,136,
    144,148,160,163,169,180,184,196,205,211,225,234,243,250,252,256,259,265,
    279,289,295,297,298,306,316,324,342,360,361,400,406,409,414,418,423,432,
    441,448,460,478,481,484,487,490,502,520,522,526,529,562,567,576,592,601,
    603,604,610,613,619,625,630,631,640,652,657,667,675,676,691,729,748,756,
    765,766,784,792,801,810,814,829,841,844,847,874,892,900,904,916,925,927,
    928,940,952,961,972,982,1000, ...
    (A number is a Smarandache pseudo-square of first kind if some permutation
     of the digits is a perfect square, including the identity permutation.)

    (Of course, all perfect squares are Smarandache pseudo-squares of first 
     kind, but not the reverse!)

     One listed all Smarandache pseudo-squares of first kind up to 1000.
 


  70)Smarandache pseudo-squares of second kind:
    10,18,40,46,52,61,63,90,94,106,108,112,136,148,160,163,180,184,205,211,234,
    243,250,252,259,265,279,295,297,298,306,316,342,360,406,409,414,418,423,
    432,448,460,478,481,487,490,502,520,522,526,562,567,592,601,603,604,610,
    613,619,630,631,640,652,657,667,675,691,748,756,765,766,792,801,810,814,
    829,844,847,874,892,904,916,925,927,928,940,952,972,982,1000, ...
    (A non-square number is a Smarandache pseudo-square of second kind if some 
     permutation of the digits is a square.)    

     One listed all Smarandache pseudo-squares of second kind up to 1000.
 


  71)Smarandache pseudo-squares of third kind:
    10,18,40,46,52,61,63,90,94,100,106,108,112,121,136,144,148,160,163,169,180,
    184,196,205,211,225,234,243,250,252,256,259,265,279,295,297,298,306,316,
    342,360,400,406,409,414,418,423,432,441,448,460,478,481,484,487,490,502,520,
    522,526,562,567,592,601,603,604,610,613,619,625,630,631,640,652,657,667,
    675,676,691,748,756,765,766,792,801,810,814,829,844,847,874,892,900,904,
    916,925,927,928,940,952,961,972,982,1000,...
    (A number is a Smarandache pseudo-square of third kind if some nontrivial
     permutation of the digits is a square.)    

    Question:  How many Smarandache pseudo-squares of third kind are square
      numbers?  (he conjectured: an infinity).
    (There are squares which are not Smarandache pseudo-squares of third kind, 
     and the reverse:
     there are Smarandache pseudo-squares of third kind which are not squares.)

     One listed all Smarandache pseudo-squares of third kind up to 1000.



  72)Smarandache pseudo-cubes of first kind:
    1,8,10,27,46,64,72,80,100,125,126,152,162,207,215,216,251,261,270,279,297,
    334,343,406,433,460,512,521,604,612,621,640,702,720,729,792,800,927,972,
    1000,...
    (A number is a Smarandache pseudo-cube of first kind if some permutation
     of the digits is a cube, including the identity permutation.)

    (Of course, all perfect cubes are Smarandache pseudo-cubes of first
     kind, but not the reverse!)

     One listed all Smarandache pseudo-cubes of first kind up to 1000.



  73)Smarandache pseudo-cubes of second kind:
    10,46,72,80,100,126,152,162,207,215,251,261,270,279,297,334,406,433,460,
    521,604,612,621,640,702,720,792,800,927,972,...
    (A non-cube number is a Smarandache pseudo-cube of second kind if some 
     permutation of the digits is a cube.)    

     One listed all Smarandache pseudo-cubes of second kind up to 1000.



  74)Smarandache pseudo-cubes of third kind:
    10,46,72,80,100,125,126,152,162,207,215,251,261,270,279,297,334,343,
    406,433,460,512,521,604,612,621,640,702,720,792,800,927,972,1000,...
    (A number is a Smarandache pseudo-cube of third kind if some nontrivial
     permutation of the digits is a cube.)    

    Question:  How many Smarandache pseudo-cubes of third kind are cubes?
      (he conjectured: an infinity).
    (There are cubes which are not Smarandache pseudo-cubes of third kind, 
     and the reverse:
     there are Smarandache pseudo-cubes of third kind which are not cubes.)

     One listed all Smarandache pseudo-cubes of third kind up to 1000.



  75)Smarandache pseudo-m-powers of first kind:
    (A number is a Smarandache pseudo-m-power of first kind if some permutation
     of the digits is an m-power, including the identity permutation; m >= 2.)


  76)Smarandache pseudo-m-powers of second kind:
    (A non m-power number is a Smarandache pseudo-m-power of second kind if 
     some permutation of the digits is an m-power; m >= 2.)

  77)Smarandache pseudo-m-powers of third kind:
    (A number is a Smarandache pseudo-m-power of third kind if some nontrivial
     permutation of the digits is an m-power; m >= 2.)    

    Question:  How many Smarandache pseudo-m-powers of third kind are m-power
      numbers?  (he conjectured: an infinity).
    (There are m-powers which are not Smarandache pseudo-m-powers of third 
     kind, and the reverse:
     there are Smarandache pseudo-m-powers of third kind which are not 
     m-powers.)


    References:
      Florentin Smarandache, "Only Problems, not Solutions!", Xiquan 
        Publishing House, Phoenix-Chicago, 1990, 1991, 1993; 
        ISBN: 1-879585-00-6.
        (reviewed in <Zentralblatt fur Mathematik> by P. Kiss: 11002, 
         pre744, 1992;
         and <The American Mathematical Monthly>, Aug.-Sept. 1991);
      Arizona State University, Hayden Library, "The Florentin Smarandache
        papers" special collection, Tempe, AZ 85287-1006, USA, phone: 
        (602)965-6515 (Carol Moore librarian), email: ICCLM@ASUACAD.BITNET . 
      "The Encyclopedia of Integer Sequences", by N. J. A. Sloane and 
        S. Plouffe, Academic Press, San Diego, New York, Boston, London, 
        Sydney, Tokyo, Toronto, 1995;
        also online, email:  superseeker@research.att.com ( SUPERSEEKER by 
        N. J. A. Sloane, S. Plouffe, B. Salvy,  ATT Bell Labs, Murray Hill,
        NJ 07974, USA); 



  78)Smarandache pseudo-factorials of first kind:
     1,2,6,10,20,24,42,60,100,102,120,200,201,204,207,210,240,270,402,420,600,
     702,720,1000,1002,1020,1200,2000,2001,2004,2007,2010,2040,2070,2100,2400,
     2700,4002,4005,4020,4050,4200,4500,5004,5040,5400,6000,7002,7020,7200,...
    (A number is a Smarandache pseudo-factorial of first kind if
     some permutation of the digits is a factorial number, including the 
     identity permutation.)    

    (Of course, all factorials are Smarandache pseudo-factorials of first kind, 
     but not the reverse!)

     One listed all Smarandache pseudo-factorials of first kind up to 10000.
     
     Procedure to obtain this sequence:
       - calculate all factorials with one digit only (1!=1, 2!=2, and 3!=6),
         this is line_1 (of one digit pseudo-factorials):
         1,2,6;
       - add 0 (zero) at the end of each element of line_1,
         calculate all factorials with two digits (4!=24 only)
         and all permutations of their digits:
         this is line_2 (of two digits pseudo-factorials):
         10,20,60; 24, 42;
       - add 0 (zero) at the end of each element of line_2 as well as anywhere
         in between their digits,
         calculate all factorials with three digits (5!=120, and 6!=720)
         and all permutations of their digits:
         this is line_3 (of three digits pseudo-factorials):
         100,200,600,240,420,204,402; 120,720, 102,210,201,702,270,720;
       and so on ...
       to get from line_k to line_(k+1) do:
       - add 0 (zero) at the end of each element of line_k as well as anywhere
         in between their digits,
         calculate all factorials with (k+1) digits
         and all permutations of their digits;
       The set will be formed by all line_1 to the last line elements
       in an increasing order.
      
       The Smarandache pseudo-factorials of second kind and third kind can
       be deduced from the first kind ones..


  79)Smarandache pseudo-factorials of second kind:
     10,20,42,60,100,102,200,201,204,207,210,240,270,402,420,600,
     702,1000,1002,1020,1200,2000,2001,2004,2007,2010,2040,2070,2100,2400,
     2700,4002,4005,4020,4050,4200,4500,5004,5400,6000,7002,7020,7200,...
    (A non-factorial number is a Smarandache pseudo-factorial of second kind if 
     some permutation of the digits is a factorial number.)    


  80)Smarandache pseudo-factorials of third kind:
     10,20,42,60,100,102,200,201,204,207,210,240,270,402,420,600,
     702,1000,1002,1020,1200,2000,2001,2004,2007,2010,2040,2070,2100,2400,
     2700,4002,4005,4020,4050,4200,4500,5004,5400,6000,7002,7020,7200,...
    (A number is a Smarandache pseudo-factorial of third kind if some nontrivial
     permutation of the digits is a factorial number.)    

    Question:  How many Smarandache pseudo-factorials of third kind are 
      factorial numbers?  (he conjectured: none! ... that means the Smarandache
      pseudo-factorials of second kind set and Smarandache pseudo-factorials of 
      third kind set coincide!).

    (Unfortunatelly, the second and third kinds of Smarandache pseudo-factorials
     coincide.)

  81)Smarandache pseudo-divisors of first kind:
    1,10,100,1,2,10,20,100,200,1,3,10,30,100,300,1,2,4,10,20,40,100,200,400,
    1,5,10,50,100,500,1,2,3,6,10,20,30,60,100,200,300,600,1,7,10,70,100,700,
    1,2,4,8,10,20,40,80,100,200,400,800,1,3,9,10,30,90,100,300,900,1,2,5,10,
    20,50,100,200,500,1000,...
    (The Smarandache pseudo-divisors of first kind of n.)

    (A number is a Smarandache pseudo-divisor of first kind of n if
     some permutation of the digits is a divisor of n, including the 
     identity permutation.)    

    (Of course, all divisors are Smarandache pseudo-divisors of first kind, 
     but not the reverse!)

     A strange property:  any integer has an infinity of Smarandache 
     pseudo-divisors of first kind !!
     because 10...0 becomes 0...01 = 1, by a circular permutation of its digits,
     and 1 divides any integer !
     
     One listed all Smarandache pseudo-divisors of first kind up to 1000
     for the numbers 1, 2, 3, ..., 10.

     Procedure to obtain this sequence:
       - calculate all divisors with one digit only,
         this is line_1 (of one digit pseudo-divisors);
       - add 0 (zero) at the end of each element of line_1,
         calculate all divisors with two digits
         and all permutations of their digits:
         this is line_2 (of two digits pseudo-divisors);
       - add 0 (zero) at the end of each element of line_2 as well as anywhere
         in between their digits,
         calculate all divisors with three digits
         and all permutations of their digits:
         this is line_3 (of three digits pseudo-divisors);
       and so on ...
       to get from line_k to line_(k+1) do:
       - add 0 (zero) at the end of each element of line_k as well as anywhere
         in between their digits,
         calculate all divisors with (k+1) digits
         and all permutations of their digits;
       The set will be formed by all line_1 to the last line elements
       in an increasing order.
      
       The Smarandache pseudo-divisors of second kind and third kind can
       be deduced from the first kind ones.


  82)Smarandache pseudo-divisors of second kind:
    10,100,10,20,100,200,10,30,100,300,10,20,40,100,200,400,10,50,100,500,10,
    20,30,60,100,200,300,600,10,70,100,700,10,20,40,80,100,200,400,800,10,30,
    90,100,300,900,20,50,100,200,500,1000,...
    (The Smarandache pseudo-divisors of second kind of n.)

    (A non-divisor of n is a Smarandache pseudo-divisor of second kind of n 
     if some permutation of the digits is a divisor of n.)    


  83)Smarandache pseudo-divisors of third kind:
    10,100,10,20,100,200,10,30,100,300,10,20,40,100,200,400,10,50,100,500,10,
    20,30,60,100,200,300,600,10,70,100,700,10,20,40,80,100,200,400,800,10,30,
    90,100,300,900,10,20,50,100,200,500,1000,...
    (The Smarandache pseudo-divisors of third kind of n.)

    (A number is a Smarandache pseudo-divisor of third kind of n if some 
     nontrivial permutation of the digits is a divisor of n.)    

     A strange property:  any integer has an infinity of Smarandache 
     pseudo-divisors of third kind !!
     because 10...0 becomes 0...01 = 1, by a circular permutation of its digits,
     and 1 divides any integer !
     
     There are divisors of n which are not Smarandache pseudo-divisors of 
     third kind of n,
     and the reverse:
     there are Smarandache pseudo-divisors of third kind of n which are not
     divisors of n.



  84)Smarandache pseudo-odd numbers of first kind:
    1,3,5,7,9,10,11,12,13,14,15,16,17,18,19,21,23,25,27,29,30,31,32,33,34,35,
    36,37,38,39,41,43,45,47,49,50,51,52,53,54,55,56,57,58,59,61,63,65,67,69,70,
    71,72,73,74,75,76,...
    (Some permutation of digits is an odd number.)


  85)Smarandache pseudo-odd numbers of second kind:
    10,12,14,16,18,30,32,34,36,38,50,52,54,56,58,70,72,74,76,78,90,92,94,96,98,
    100,102,104,106,108,110,112,114,116,118,...
    (Even numbers such that some permutation of digits is an odd number.)


  86)Smarandache pseudo-odd numbers of third kind:
    10,11,12,13,14,15,16,17,18,19,30,31,32,33,34,35,36,37,38,39,50,51,52,53,54,
    55,56,57,58,59,70,71,72,73,74,75,76,...
    (Nontrivial permutation of digits is an odd number.)


  87)Smarandache pseudo-triangular numbers:
    1,3,6,10,12,15,19,21,28,30,36,45,54,55,60,61,63,66,78,82,87,91,...
    (Some permutation of digits is a triangular number.)
    
    A triangular number has the general form:  n(n+1)/2.


  88)Smarandache pseudo-even numbers of first kind:
    0,2,4,6,8,10,12,14,16,18,20,21,22,23,24,25,26,27,28,29,30,32,34,36,38,40,
    41,42,43,44,45,46,47,48,49,50,52,54,56,58,60,61,62,63,64,65,66,67,68,69,70,
    72,74,76,78,80,81,82,83,84,85,86,87,88,89,90,92,94,96,98,100,...
    (The Smarandache pseudo-even numbers of first kind.)

    (A number is a Smarandache pseudo-even number of first kind if
     some permutation of the digits is a even number, including the 
     identity permutation.)    

    (Of course, all even numbers are Smarandache pseudo-even numbers of first 
     kind, but not the reverse!)

     A strange property:  an odd number can be a Smarandache pseudo-even 
     number!
     
     One listed all Smarandache pseudo-even numbers of first kind up to 100.


  89)Smarandache pseudo-even numbers of second kind:
    21,23,25,27,29,41,43,45,47,49,61,63,65,67,69,81,83,85,87,89,101,103,105,
    107,109,121,123,125,127,129,141,143,145,147,149,161,163,165,167,169,181,
    183,185,187,189,201,...
    (The Smarandache pseudo-even numbers of second kind.)

    (A non-even number is a Smarandache pseudo-even number of second kind 
     if some permutation of the digits is a even number.)    


  90)Smarandache pseudo-even numbers of third kind:
    20,21,22,23,24,25,26,27,28,29,40,41,42,43,44,45,46,47,48,49,60,61,62,63,64,
    65,66,67,68,69,80,81,82,83,84,85,86,87,88,89,100,101,102,103.104,105,106,
    107,108,109,110,120,121,122,123,124,125,126,127,128,129,130,...
    (The Smarandache pseudo-even numbers of third kind.)

    (A number is a Smarandache pseudo-even number of third kind if some 
     nontrivial permutation of the digits is a even number.)    


  91)Smarandache pseudo-multiples of first kind (of 5):
    0,5,10,15,20,25,30,35,40,45,50,51,52,53,54,55,56,57,58,59,60,65,70,75,80,
    85,90,95,100,101,102,103,104,105,106,107,108,109,110,115,120,125,130,135,
    140,145,150,151,152,153,154,155,156,157,158,159,160,165,...
    (The Smarandache pseudo-multiples of first kind of 5.)

    (A number is a Smarandache pseudo-multiple of first kind of 5 if
     some permutation of the digits is a multiple of 5, including the 
     identity permutation.)    

    (Of course, all multiples of 5 are Smarandache pseudo-multiples of first 
     kind, but not the reverse!)


  92)Smarandache pseudo-multiples of second kind (of 5):
    51,52,53,54,56,57,58,59,101,102,103,104,106,107,108,109,151,152,153,154,
    156,157,158,159,201,202,203,204,206,207,208,209,251,252,253,254,256,257,
    258,259,301,302,303,304,306,307,308,309,351,352...
    (The Smarandache pseudo-multiples of second kind of 5.)

    (A non-multiple of 5 is a Smarandache pseudo-multiple of second kind of 5 
     if some permutation of the digits is a multiple of 5.)    


  93)Smarandache pseudo-multiples of third kind (of 5):
    50,51,52,53,54,55,56,57,58,59,100,101,102,103,104,105,106,107,108,109,110,
    115,120,125,130,135,140,145,150,151,152,153,154,155,156,157,158,159,160,
    165,170,175,180,185,190,195,200,...
    (The Smarandache pseudo-multiples of third kind of 5.)

    (A number is a Smarandache pseudo-multiple of third kind of 5 if some 
     nontrivial permutation of the digits is a multiple of 5.)    


  94)Smarandache pseudo-multiples of first kind of p (p is an integer >= 2)
   {Generalizations}:
    (The Smarandache pseudo-multiples of first kind of p.)

    (A number is a Smarandache pseudo-multiple of first kind of p if
     some permutation of the digits is a multiple of p, including the 
     identity permutation.)    

    (Of course, all multiples of p are Smarandache pseudo-multiples of first 
     kind, but not the reverse!)


     Procedure to obtain this sequence:
       - calculate all multiples of p with one digit only (if any),
         this is line_1 (of one digit pseudo-multiples of p);
       - add 0 (zero) at the end of each element of line_1,
         calculate all multiples of p with two digits (if any)
         and all permutations of their digits:
         this is line_2 (of two digits pseudo-multiples of p);
       - add 0 (zero) at the end of each element of line_2 as well as anywhere
         in between their digits,
         calculate all multiples with three digits (if any)
         and all permutations of their digits:
         this is line_3 (of three digits pseudo-multiples of p);
       and so on ...
       to get from line_k to line_(k+1) do:
       - add 0 (zero) at the end of each element of line_k as well as anywhere
         in between their digits,
         calculate all multiples with (k+1) digits (if any)
         and all permutations of their digits;
       The set will be formed by all line_1 to the last line elements
       in an increasing order.
      
       The Smarandache pseudo-multiples of second kind and third kind of p can
       be deduced from the first kind ones.


  95)Smarandache pseudo-multiples of second kind of p (p is an integer >= 2):
    (The Smarandache pseudo-multiples of second kind of p.)

    (A non-multiple of p is a Smarandache pseudo-multiple of second kind of p 
     if some permutation of the digits is a multiple of p.)    


  96)Smarandache pseudo-multiples of third kind of p (p is an integer >= 2):
    (The Smarandache pseudo-multiples of third kind of p.)

    (A number is a Smarandache pseudo-multiple of third kind of p if some 
     nontrivial permutation of the digits is a multiple of p.)    



    References:
      Florentin Smarandache, "Only Problems, not Solutions!", Xiquan 
        Publishing House, Phoenix-Chicago, 1990, 1991, 1993; 
        ISBN: 1-879585-00-6.
        (reviewed in <Zentralblatt fur Mathematik> by P. Kiss: 11002, 
         pre744, 1992;
         and <The American Mathematical Monthly>, Aug.-Sept. 1991);
      Arizona State University, Hayden Library, "The Florentin Smarandache
        papers" special collection, Tempe, AZ 85287-1006, USA, phone: 
        (602)965-6515 (Carol Moore librarian), email: ICCLM@ASUACAD.BITNET . 



  97)Smarandache constructive set (of digits 1,2):
    1,2,11,12,21,22,111,112,121,122,211,212,221,222,1111,1112,1121,1122,1211,
    1212,1221,1222,21112112,2121,2122,2211,2212,2221,2222,...
    (Numbers formed by digits 1 and 2 only.)

    Definition:
      a1) 1, 2 belong to S;
                                    __
      a2) if a, b belong to S, then ab belongs to S too; 
      a3) only elements obtained by rules a1) and a2) applied a finite number
            of times belong to S.

    Remark:
      - there are 2^k numbers of k digits in the sequence, for k = 1, 2, 
          3, ... ;
      - to obtain from the k-digits number group the (k+1)-digits number 
          group, just put first the digit 1 and second the digit 2 in the
          front of all k-digits numbers.




  98)Smarandache constructive set (of digits 1,2,3):
    1,2,3,11,12,13,21,22,23,31,32,33,111,112,113,121,122,123,131,132,133,211,
    212,213,221,222,223,231,232,233,311,312,313,321,322,323,331,332,333,...
    (Numbers formed by digits 1, 2, and 3 only.)

    Definition:
      a1) 1, 2, 3 belong to S;
                                    __
      a2) if a, b belong to S, then ab belongs to S too; 
      a3) only elements obtained by rules a1) and a2) applied a finite number
            of times belong to S.

    Remark:
      - there are 3^k numbers of k digits in the sequence, for k = 1, 2, 
          3, ... ;
      - to obtain from the k-digits number group the (k+1)-digits number 
          group, just put first the digit 1, second the digit 2, and third
          the digit 3 in the front of all k-digits numbers.



  99)Smarandache generalizated constructive set:
    (Numbers formed by digits d , d , ..., d  only, 
                               1   2        m
     all d  being different each other, 1 <= m <= 9.)
          i

    Definition:
      a1) d , d , ..., d  belong to S;
           1   2        m
                                    __
      a2) if a, b belong to S, then ab belongs to S too; 
      a3) only elements obtained by rules a1) and a2) applied a finite number
            of times belong to S.

    Remark:
      - there are m^k numbers of k digits in the sequence, for k = 1, 2, 
          3, ... ;
      - to obtain from the k-digits number group the (k+1)-digits number 
          group, just put first the digit d , second the digit d , ..., and 
                                           1                    2
          the m-th time digit d  in the front of all k-digits numbers.
                               m

    More general:  all digits d  can be replaced by numbers as large as we want
                               i
    (therefore of many digits each), and also m can be as large as we want.



  100)Smarandache square roots:
    0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,
    6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,
    8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,
    10,10,10,10,10,10,10,10,10,10,10,10,10,10,...
    (s (n) is the superior integer part of square root of n.)
      q

    Remark:  this sequence is the natural sequence, where each number is 
      repeated 2n+1 times,
      because between n^2 (included) and (n+1)^2 (excluded) there are
      (n+1)^2 - n^2 different numbers.



  101)Smarandache cubical roots:
    0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,
    3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,
    4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,
    4,4,4,4,4,4,4,4,4,4,4,4,4,4,...
    (c (n) is the superior integer part of cubical root of n.)
      q

    Remark:  this sequence is the natural sequence, where each number is 
      repeated 3n^2 + 3n + 1 times,
      because between n^3 (included) and (n+1)^3 (excluded) there are
      (n+1)^3 - n^3 different numbers.



  102)Smarandache m-power roots:
    (m (n) is the superior integer part of m-power root of n.)
      q

    Remark:  this sequence is the natural sequence, where each number is 
      repeated (n+1)^m - n^m times.



  103)Smarandache numerical carpet:
      has the general form



                                    .
                                    .
                                    .
                                    1
                                   1a1
                                  1aba1
                                 1abcba1
                                1abcdcba1
                               1abcdedcba1
                              1abcdefedcba1
                          ...1abcdefgfedcba1...
                              1abcdefedcba1
                               1abcdedcba1
                                1abcdcba1
                                 1abcba1
                                  1aba1
                                   1a1
                                    1
                                    .
                                    .
                                    .


      On the border of level 0, the elements are equal to "1";  
        they form a rhomb.
      Next, on the border of level 1, the elements are equal to "a",
        where "a" is the sum of all elements of the previous border;
        the "a"s form a rhomb too inside the previous one.  
      Next again, on the border of level 2, the elements are equal to "b",
        where "b" is the sum of all elements of the previous border;
        the "b"s form a rhomb too inside the previous one.  
      And so on...
      The Smarandache carpet is symmetric and esthetic, in its middle g is the 
      sum of all carpet numbers (the core).

      Look at a few terms of the Smarandache Numerical Carpet:

                                    1

                                    1
                                   141
                                    1

                                    1
                                 1  8  1
                              1  8 40  8  1
                                 1  8  1
                                    1

                                    1
                                1  12   1
                            1  12 108  12  1
                         1 12 108 540 108 12 1       
                            1  12 108  12  1
                                1  12   1
                                    1

                                    1
                               1   16    1
                          1   16  208   16   1
                      1  16  208 1872  208  16  1
                   1 16 208 1872 9360 1872 208 16 1
                       1 16  208 1872  208  16  1
                          1   16  208   16   1
                               1   16    1
                                    1

                                    1
                             1     20     1
                       1    20    340    20    1
                  1   20   340   4420   340   20   1
              1  20  340  4420  39780  4420  340  20  1
           1 20 340 4420 39780 198900 39780 4420 340 20 1
              1  20  340  4420  39780  4420  340  20  1
                  1   20   340   4420   340   20   1
                       1    20    340    20    1
                             1     20     1
                                    1     

                                    .
                                    .
                                    .


      Or, under other form:
            1
            1  4
            1  8  40
            1 12 108   504
            1 16 208  1872   9360
            1 20 340  4420  39780  198900 
            1 24 504  8568 111384 1002456   5012280
            1 28 700 14700 249900 3248700  29238300 146191500
            1 32 928 23200 487200 8282400 107671200 969040800 4845204000
            ..................................................................
            .
            .
            .



      General Formula:

                    k
                  _____
       C(n,k) = 4n|   |(4n-4i+1) for 1 <= k <= n, 
                   i=1                         

       and C(n,0) = 1.



     References:
      Arizona State University, Hayden Library, "The Florentin Smarandache
        papers" special collection, Tempe, AZ 85287-1006, USA, phone: 
        (602)965-6515 (Carol Moore librarian), email: ICCLM@ASUACAD.BITNET . 
      Student Conference, University of Craiova, Department of Mathematics,
        April 1979,  "Some problems in number theory" by Florentin Smarandache.
      Fl. Smarandache, "Collected Papers" (Vol. 1), Ed. Tempus, Bucharest, 
        1995;      


  104)Goldbach-Smarandache table:
       6,10,14,18,26,30,38,42,42,54,62,74,74,90,...
       (t(n) is the largest even number such that any other even number not 
        exceeding it is the sum of two of the first n odd primes.)
     
     It helps to better understand Goldbach's conjecture:
       - if t(n) is unlimited, then the conjecture is true;
       - if t(n) is constant after a certain rank, then the conjecture is false.

     Also, the table gives how many times an even number is written as a sum of
     two odd primes, and in what combinations -- which can be found in the
     "Encyclopedia of Integer Sequences" by N. J. A. Sloane and S. Plouffe,
     Academic Press, San Diego, New York, Boston, London, Sydney, Tokyo, 
     Toronto, 1995.

     Of course, t(n) <= 2p  , where p  is the n-th odd prime, n = 1, 2, 3, ... .
                          n          n

     Here is the table:

               +   3  5  7 11 13 17 19 23 29 31 37 41 43 47 
                  ------------------------------------------ . . .
               3 | 6  8 10 14 16 20 22 26 32 34 40 44 46 50 .
               5 |   10 12 16 18 22 24 28 34 36 42 46 48 52 .
               7 |      14 18 20 24 26 30 36 38 44 48 50 54 .
              11 |         22 24 28 30 34 40 42 48 52 54 58 .
              13 |            26 30 32 36 42 44 50 54 56 60 .
              17 |               34 36 40 46 48 54 58 60 64 .
              19 |                  38 42 48 50 56 60 62 66 .
              23 |                     46 52 54 60 64 66 70 .
              29 |                        58 60 66 70 72 76 .
              31 |                           62 68 72 74 78 .
              37 |                              74 78 80 84 .
              41 |                                 82 84 88 .
              43 |                                    86 90 .
              47 |                                       94 .
                 ............................................
                 .                                            .
                 .                                              .
                 .                                                .




  105)Smarandache-Vinogradov table:
     9,15,21,29,39,47,57,65,71,93,99,115,129,137,...
     (v(n) is the largest odd number such that any odd number >= 9 not 
      exceeding it is the sum of three of the first n odd primes.)

     It helps to better understand Goldbach's conjecture for three primes:
       - if v(n) is unlimited, then the conjecture is true;
       - if v(n) is constant after a certain rank, then the conjecture is false.
      (Vinogradov proved in 1937 that any odd number greater than 3^(3^15) 
      satisfies this conjecture.  
      But what about values less than 3^(3^15) ?)

     Also, the table gives you in how many different combinations an odd number
     is written as a sum of three odd primes, and in what combinations.

     Of course, v(n) <= 3p  , where p  is the n-th odd prime, n = 1, 2, 3, ... .
                          n          n
     It is also generalized for the sum of m primes,
     and how many times a number is written as a sum of m primes (m > 2).


     This is a 3-dimensional 14x14x14 table, that we can expose only as 14 
     planar 14x14 tables (using Goldbach-Smarandache table):

            -----
           | 3   |
           |  +  |
           |     | 3  5  7 11 13 17 19 23 29 31 37 41 43 47 
            ----- ------------------------------------------ . . .
               3 | 9 11 13 17 19 23 25 29 35 37 43 47 49 53 .
               5 |   13 15 19 21 25 27 31 37 39 45 49 51 55 .
               7 |      17 21 23 27 29 33 39 41 47 51 53 57 .
              11 |         25 27 31 33 37 43 45 51 55 57 61 .
              13 |            29 33 35 39 45 47 53 57 59 63 .
              17 |               37 39 43 49 51 57 61 63 67 .
              19 |                  41 45 51 53 59 63 65 69 .
              23 |                     49 55 57 63 67 69 73 .
              29 |                        61 63 69 73 75 79 .
              31 |                           65 71 75 77 81 .
              37 |                              77 81 83 87 .
              41 |                                 85 87 91 .
              43 |                                    89 93 .
              47 |                                       97 .
                 ............................................
                 .                                            .
                 .                                              .
                 .                                                .


            -----
           | 5   |
           |  +  |
           |     | 3  5  7 11 13 17 19 23 29 31 37 41 43 47 
            ----- ------------------------------------------ . . .
               3 |11 13 15 19 21 25 27 31 37 39 45 49 51 55 .
               5 |   15 17 21 23 27 29 33 39 41 47 51 53 57 .
               7 |      19 23 25 29 31 35 41 43 49 53 55 59 .
              11 |         27 29 33 35 39 45 47 53 57 59 63 .
              13 |            31 35 37 41 47 49 55 59 61 65 .
              17 |               39 41 45 51 53 59 63 65 69 .
              19 |                  43 47 53 55 61 65 67 71 .
              23 |                     51 57 59 65 69 71 75 .
              29 |                        63 65 71 75 77 81 .
              31 |                           67 73 77 79 83 .
              37 |                              79 83 85 89 .
              41 |                                 87 89 93 .
              43 |                                    91 95 .
              47 |                                       99 .
                 ............................................
                 .                                            .
                 .                                              .
                 .                                                .


            -----
           | 7   |
           |  +  |
           |     | 3  5  7 11 13 17 19 23 29 31 37 41 43  47 
            ----- ------------------------------------------- . . .
               3 |13 15 17 21 23 27 29 33 39 41 47 51 53  57 .
               5 |   17 19 23 25 29 31 35 41 43 49 53 55  59 .
               7 |      21 25 27 31 33 37 43 45 51 55 57  61 .
              11 |         29 31 35 37 41 47 49 55 59 61  65 .
              13 |            33 37 39 43 49 51 57 61 63  67 .
              17 |               41 43 47 53 55 61 65 67  71 .
              19 |                  45 49 55 57 63 67 69  73 .
              23 |                     53 59 61 67 71 73  77 .
              29 |                        65 67 73 77 79  83 .
              31 |                           69 75 79 81  85 .
              37 |                              81 85 87  91 .
              41 |                                 89 91  95 .
              43 |                                    93  97 .
              47 |                                       101 .
                 .............................................
                 .                                             .
                 .                                               .
                 .                                                 .


            -----
           |11   |
           |  +  |
           |     | 3  5  7 11 13 17 19 23 29 31 37 41 43  47 
            ----- ------------------------------------------- . . .
               3 |17 19 21 25 27 31 33 37 43 45 51 55 57  61 .
               5 |   21 23 27 29 33 35 39 45 47 53 57 59  63 .
               7 |      25 29 31 35 37 41 47 49 55 59 61  65 .
              11 |         33 35 39 41 45 51 53 59 63 65  69 .
              13 |            37 41 43 47 53 55 61 65 67  71 .
              17 |               45 47 51 57 59 65 69 71  75 .
              19 |                  49 53 59 61 67 71 73  77 .
              23 |                     57 63 65 71 75 77  81 .
              29 |                        69 71 77 81 83  87 .
              31 |                           73 79 83 85  89 .
              37 |                              85 89 91  95 .
              41 |                                 93 95  99 .
              43 |                                    97 101 .
              47 |                                       105 .
                 .............................................
                 .                                             .
                 .                                               .
                 .                                                 .


            -----
           |13   |
           |  +  |
           |     | 3  5  7 11 13 17 19 23 29 31 37 41 43  47 
            ----- ------------------------------------------- . . .
               3 |19 21 23 27 29 33 35 39 45 47 53 57 59  63 .
               5 |   23 25 29 31 35 37 41 47 49 55 59 61  65 .
               7 |      27 31 33 37 39 43 49 51 57 61 63  67 .
              11 |         35 37 41 43 47 53 55 61 65 67  71 .
              13 |            39 43 45 49 55 57 63 67 69  73 .
              17 |               47 49 53 59 61 67 71 73  77 .
              19 |                  51 55 61 63 69 73 75  79 .
              23 |                     59 65 67 73 77 79  83 .
              29 |                        71 73 79 83 85  89 .
              31 |                           75 81 85 87  91 .
              37 |                              87 91 93  97 .
              41 |                                 95 97 101 .
              43 |                                    99 103 .
              47 |                                       107 .
                 .............................................
                 .                                             .
                 .                                               .
                 .                                                 .

            -----
           |17   |
           |  +  |
           |     | 3  5  7 11 13 17 19 23 29 31 37 41  43  47 
            ----- ------------------------------------------- . . .
               3 |23 25 27 31 33 37 39 43 49 51 57 61  63  67 .
               5 |   27 29 33 35 39 41 45 51 53 59 63  65  69 .
               7 |      31 35 37 41 43 47 53 55 61 65  67  71 .
              11 |         39 41 45 47 51 57 59 65 69  71  75 .
              13 |            43 47 49 53 59 61 67 71  73  77 .
              17 |               51 53 57 63 65 71 75  77  81 .
              19 |                  55 59 65 67 73 77  79  83 .
              23 |                     63 69 71 77 81  83  87 .
              29 |                        75 77 83 87  89  93 .
              31 |                           79 85 89  91  95 .
              37 |                              91 95  97 101 .
              41 |                                 99 101 105 .
              43 |                                    103 107 .
              47 |                                        111 .
                 .............................................
                 .                                             .
                 .                                               .
                 .                                                 .


            -----
           |19   |
           |  +  |
           |     | 3  5  7 11 13 17 19 23 29 31 37  41  43  47 
            ----- -------------------------------------------- . . .
               3 |25 27 29 33 35 39 41 45 51 53 59  63  65  69 .
               5 |   29 31 35 37 41 43 47 53 55 61  65  67  71 .
               7 |      33 37 39 43 45 49 55 57 63  67  69  73 .
              11 |         41 43 47 49 53 59 61 67  71  73  77 .
              13 |            45 49 51 55 61 63 69  73  75  79 .
              17 |               53 55 59 65 67 73  77  79  83 .
              19 |                  57 61 67 69 75  79  81  85 .
              23 |                     65 71 73 79  83  85  89 .
              29 |                        77 79 85  89  91  95 .
              31 |                           81 87  91  93  97 .
              37 |                              93  97  99 103 .
              41 |                                 101 103 107 .
              43 |                                     105 109 .
              47 |                                         113 .
                 ...............................................
                 .                                               .
                 .                                                 .
                 .                                                   .

            -----
           |23   |
           |  +  |
           |     | 3  5  7 11 13 17 19 23 29 31 37  41  43  47 
            ----- -------------------------------------------- . . .
               3 |29 31 33 37 39 43 45 49 55 57 63  67  69  73 .
               5 |   33 35 39 41 45 47 51 57 59 65  69  71  75 .
               7 |      37 41 43 47 49 53 59 61 67  71  73  77 .
              11 |         45 47 51 53 57 63 65 71  75  77  81 .
              13 |            49 53 55 59 65 67 73  77  79  83 .
              17 |               57 59 63 69 71 77  81  83  87 .
              19 |                  61 65 71 73 79  83  85  89 .
              23 |                     69 75 77 83  87  89  93 .
              29 |                        81 83 89  93  95  99 .
              31 |                           85 91  95  97 101 .
              37 |                              97 101 103 107 .
              41 |                                 105 107 111 .
              43 |                                     109 113 .
              47 |                                         117 .
                 ...............................................
                 .                                               .
                 .                                                 .
                 .                                                   .


            -----
           |29   |
           |  +  |
           |     | 3  5  7 11 13 17 19 23 29 31  37  41  43  47 
            ----- --------------------------------------------- . . .
               3 |35 37 39 43 45 49 51 55 61 63  69  73  75  79 .
               5 |   39 41 45 47 51 53 57 63 65  71  75  77  81 .
               7 |      43 47 49 53 55 59 65 67  73  77  79  83 .
              11 |         51 53 57 59 63 69 71  77  81  83  87 .
              13 |            55 59 61 65 71 73  79  83  85  89 .
              17 |               63 65 69 75 77  83  87  89  93 .
              19 |                  67 71 77 79  85  89  91  95 .
              23 |                     75 81 83  89  93  95  99 .
              29 |                        87 89  95  99 101 105 .
              31 |                           91  97 101 103 107 .
              37 |                              103 107 109 113 .
              41 |                                  111 113 117 .
              43 |                                      115 119 .
              47 |                                          123 .
                 ................................................
                 .                                                .
                 .                                                  .
                 .                                                    .


            -----
           |31   |
           |  +  |
           |     | 3  5  7 11 13 17 19 23 29 31  37  41  43  47 
            ----- --------------------------------------------- . . .
               3 |37 39 41 45 47 51 53 57 63 65  71  75  77  81 .
               5 |   41 43 47 49 53 55 59 65 67  73  77  79  83 .
               7 |      45 49 51 55 57 61 67 69  75  79  81  85 .
              11 |         53 55 59 61 65 71 73  79  83  85  89 .
              13 |            57 61 63 67 73 75  81  85  87  91 .
              17 |               65 67 71 77 79  85  89  91  95 .
              19 |                  69 73 79 81  87  91  93  97 .
              23 |                     77 83 85  91  95  97 101 .
              29 |                        89 91  97 101 103 107 .
              31 |                           93  99 103 105 109 .
              37 |                              105 109 111 115 .
              41 |                                  113 115 119 .
              43 |                                      117 121 .
              47 |                                          125 .
                 ................................................
                 .                                                .
                 .                                                  .
                 .                                                    .



            -----
           |37   |
           |  +  |
           |     | 3  5  7 11 13 17 19 23 29 31  37  41  43  47 
            ----- --------------------------------------------- . . .
               3 |43 45 47 51 53 57 59 63 69 71  77  81  83  87 .
               5 |   47 49 53 55 59 61 65 71 73  79  83  85  89 .
               7 |      51 55 57 61 63 67 73 75  81  85  87  91 .
              11 |         59 61 65 67 71 77 79  85  89  91  95 .
              13 |            63 67 69 73 79 81  87  91  93  97 .
              17 |               71 73 77 83 85  91  95  97 101 .
              19 |                  75 79 85 87  93  97  99 103 .
              23 |                     83 89 91  97 101 103 107 .
              29 |                        95 97 103 107 109 113 .
              31 |                           99 105 109 111 115 .
              37 |                              111 115 117 121 .
              41 |                                  119 121 125 .
              43 |                                      123 127 .
              47 |                                          131 .
                 ................................................
                 .                                                .
                 .                                                  .
                 .                                                    .



            -----
           |41   |
           |  +  |
           |     | 3  5  7 11 13 17 19 23 29  31  37  41  43  47 
            ----- ---------------------------------------------- . . .
               3 |47 49 51 55 57 61 63 67 73  75  81  85  87  91 .
               5 |   51 53 57 59 63 65 69 75  77  83  87  89  93 .
               7 |      55 59 61 65 67 71 77  79  85  89  91  95 .
              11 |         63 65 69 71 75 81  83  89  93  95  99 .
              13 |            67 71 73 77 83  85  91  95  97 101 .
              17 |               75 77 81 87  89  95  99 101 105 .
              19 |                  79 83 89  91  97 101 103 107 .
              23 |                     87 93  95 101 105 107 111 .
              29 |                        99 101 107 111 113 117 .
              31 |                           103 109 113 115 119 .
              37 |                               115 119 121 125 .
              41 |                                   123 125 129 .
              43 |                                       127 131 .
              47 |                                           135 .
                 .................................................
                 .                                                 .
                 .                                                   .
                 .                                                     .


            -----
           |43   |
           |  +  |
           |     | 3  5  7 11 13 17 19 23  29  31  37  41  43  47 
            ----- ----------------------------------------------- . . .
               3 |49 51 53 57 59 63 65 69  75  77  83  87  89  93 .
               5 |   53 55 59 61 65 67 71  77  79  85  89  91  95 .
               7 |      57 61 63 67 69 73  79  81  87  91  93  97 .
              11 |         65 67 71 73 77  83  85  91  95  97 101 .
              13 |            69 73 75 79  85  87  93  97  99 103 .
              17 |               77 79 83  89  91  97 101 103 107 .
              19 |                  81 85  91  93  99 103 105 109 .
              23 |                     89  95  97 103 107 109 113 .
              29 |                        101 103 109 113 115 119 .
              31 |                            105 111 115 117 121 .
              37 |                                117 121 123 127 .
              41 |                                    125 127 131 .
              43 |                                        129 133 .
              47 |                                            137 .
                 ..................................................
                 .                                                  .
                 .                                                    .
                 .                                                      .



            -----
           |47   |
           |  +  |
           |     | 3  5  7 11 13 17 19 23  29  31  37  41  43  47 
            ----- ----------------------------------------------- . . .
               3 |53 55 57 61 63 67 69 73  79  81  87  91  93  97 .
               5 |   57 59 63 65 69 71 75  81  83  89  93  95  99 .
               7 |      61 65 67 71 73 77  83  85  91  95  97 101 .
              11 |         69 71 75 77 81  87  89  95  99 101 105 .
              13 |            73 77 79 83  89  91  97 101 103 107 .
              17 |               81 83 87  93  95 101 105 107 111 .
              19 |                  85 89  95  97 103 107 109 113 .
              23 |                     93  99 101 107 111 113 117 .
              29 |                        105 107 113 117 119 123 .
              31 |                            109 115 119 121 125 .
              37 |                                121 125 127 131 .
              41 |                                    129 131 135 .
              43 |                                        133 137 .
              47 |                                            141 .
                 ..................................................
                 .                                                  .
                 .                                                    .
                 .                                                      .




  106)Smarandache-Vinogradov sequence:
     0,0,0,0,1,2,4,4,6,7,9,10,11,15,17,16,19,19,23,25,26,26,28,33,32,35,43,39,
     40,43,43,...
     (a(2k+1) represents the number of different combinations such that 2k+1
      is written as a sum of three odd primes.)

     This sequence is deduced from the Smarandache-Vinogradov table.


    References:
      Florentin Smarandache, "Only Problems, not Solutions!", Xiquan 
        Publishing House, Phoenix-Chicago, 1990, 1991, 1993; 
        ISBN: 1-879585-00-6.
        (reviewed in <Zentralblatt fur Mathematik> by P. Kiss: 11002, 
         pre744, 1992;
         and <The American Mathematical Monthly>, Aug.-Sept. 1991);
      Florentin Smarandache, "Problems with and without ... problems!", Ed.
        Somipress, Fes, Morocco, 1983;
      Arizona State University, Hayden Library, "The Florentin Smarandache
        papers" special collection, Tempe, AZ 85287-1006, USA, phone: 
        (602)965-6515 (Carol Moore librarian), email: ICCLM@ASUACAD.BITNET ;
      N. J. A. Sloane, e-mail to R. Muller, February 26, 1994.



  107)Smarandache paradoxist numbers:
     There exist a few "Smarandache" number sequences.
     A number n is called a "Smarandache paradoxist number" if and only if  
     n doesn't belong to any of the Smarandache defined numbers.
     Question:  find the Smarandache paradoxist number sequence.

   Solution?
     If a number k is a Smarandache paradoxist number, then k doesn't belong to 
   any of the Smarandache defined numbers, 
   therefore k doesn't belong to the Smarandache paradoxist numbers too!
     If a number k doesn't belong to any of the Smarandache defined numbers,
   then k is a Smarandache paradoxist number,
   therefore k belongs to a Smarandache defined numbers (because Smarandache 
   paradoxist numbers is also in the same category) -- contradiction.

     Dilemma:  is the Smarandache paradoxist number sequence empty ??



  108)Non-Smarandache numbers:
       A number n is called a "non-Smarandache number" if and only if
     n is neither a Smarandache paradoxist number nor any of the
     Smarandache defined numbers. 
       Question:  find the non-Smarandache number sequence.

       Dilemma 1:  is the non-Smarandache number sequence empty, too ??
       Dilemma 2:  is a non-Smarandache number equivalent to a Smarandache 
     paradoxist number ??? (this would be another paradox !! ... because
     a non-Smarandache number is not a Smarandache paradoxist number).


  109)The paradox of Smarandache numbers:
      Any number is a Smarandache number, the non-Smarandache number too.
      (This is deduced from the following paradox (see the reference):
         "All is possible, the impossible too!")


   Reference:
     Charles T. Le, "The Smarandache Class of Paradoxes", in <Bulletin of Pure
       and Applied Sciences>, Bombay, India, 1995;
       and in <Abracadabra>, Salinas, CA, 1993, and in <Tempus>, Bucharest, No. 
       2, 1994.




  110)Smarandache multiplication:
     Another way to multiply two integer numbers, A and B:
       - let k be an integer >= 2;
       - write A and B on two different vertical columns: c(A), respectively 
           c(B);
       - multiply A by k, and write the product A  on the column c(A);
                                                 1
       - divide B by k, and write the integer part of the quotient B
                                                                    1 
           on the column c(B);
       ... and so on with the new numbers A  and  B , 
                                           1       1
       untill we get a B  < k on the column c(B);
                        i
       Then:
       - write another column c(r), on the right side of c(B), such that:
           for each number of column c(B), which may be a multiple of k plus 
           the rest r (where r = 0, 1, 2, ..., k-1),
           the corresponding number on c(r) will be r;
       - multiply each number of column A by its corresponding r of c(r),
           and put the new products on another column c(P) on the right side 
           of c(r);
       - finally add all numbers of column c(P).
       AxB = the sum of all numbers of c(P).

      Remark that any multiplication of integer numbers can be done 
        only by multiplication with 2, 3, ..., k, divisions by k, and additions.


   This is a generalization of Russian multiplication (when k=2).

   Smarandache multiplication is usefull when k is very small, the best values
   being for k=2 (Russian multiplication -- known since Egyptian time), or k=3. 
   If k is greater than or equal to min {10, B}, this multiplication is trivial
   (the obvious multiplication).


   Example 1 (if we choose k=3):
      73x97= ? 

                                     x3  |  /3  |           
                                  -------|------|-------------
                                    c(A) | c(B) | c(r) | c(P)
                                  -------|------|------|------
                                     73  |  97  |   1  |   73 
                                    219  |  32  |   2  |  438
                                    657  |  10  |   1  |  657
                                   1971  |   3  |   0  |    0
                                   5913  |   1  |   1  | 5913
                                  ---------------------|------
                                                       | 7081 total

      therefore: 73x97=7081.
      Remark that any multiplication of integer numbers can be done
        only by multiplication with 2, 3, divisions by 3, and additions.



   Example 2 (if we choose k=4):
      73x97= ? 

                                     x4  |  /4  |           
                                  -------|------|-------------
                                    c(A) | c(B) | c(r) | c(P)
                                  -------|------|------|------
                                     73  |  97  |   1  |   73 
                                    292  |  24  |   0  |    0
                                   1168  |   6  |   2  | 2336
                                   4672  |   1  |   1  | 4672
                                  ---------------------|------
                                                       | 7081 total

      therefore: 73x97=7081.
      Remark that any multiplication of integer numbers can be done
        only by multiplication with 2, 3, 4, divisions by 4, and additions.



   Example 3 (if we choose k=5):
      73x97= ? 

                                     x5  |  /5  |           
                                  -------|------|-------------
                                    c(A) | c(B) | c(r) | c(P)
                                  -------|------|------|------
                                     73  |  97  |   2  |  146 
                                    365  |  19  |   4  | 1460
                                   1825  |   3  |   3  | 5475
                                  ---------------------|------
                                                       | 7081 total

      therefore: 73x97=7081.
      Remark that any multiplication of integer numbers can be done
        only by multiplication with 2, 3, 4, 5, divisions by 5, and additions.
 
    The Smarandache multiplication becomes less usefull when k increases.
    Look at another example (4), what happens when k=10:
      73x97= ? 

                                    x10  | /10  |           
                                  -------|------|-------------
                                    c(A) | c(B) | c(r) | c(P)
                                  -------|------|------|------
                                     73  |  97  |   7  |  511 (=73x7)   
                                    730  |   9  |   9  | 6570 (=730x9)
                                  ---------------------|------
                                                       | 7081 total

      therefore: 73x97=7081.
      Remark that any multiplication of integer numbers can be done
        only by multiplication with 2, 3, ..., 9, 10, divisions by 10, and 
        additions --
        hence we obtain just the obvious multiplication!
 


  111)Smarandache division by k^n:
     Another way to divide an integer numbers A by k^n, where k, n are 
     integers >= 2:
       - write A and k^n on two different vertical columns: c(A), respectively 
           c(k^n);
       - divide A by k, and write the integer quotient A  on the column c(A);
                                                        1
       - divide k^n by k, and write the quotient q  = k^(n-1)
                                                  1
           on the column c(k^n);

       ... and so on with the new numbers A  and  q , 
                                           1       1
       untill we get q  = 1 (= k^0) on the column c(k^n);
                      n
       Then:
       - write another column c(r), on the left side of c(A), such that:
           for each number of column c(A), which may be a multiple of k plus 
           the rest r (where r = 0, 1, 2, ..., k-1),
           the corresponding number on c(r) will be r;
       - write another column c(P), on the left side of c(r), in the following
           way: the element on line i (except the last line which is 0) will 
           be k^(i-1);
       - multiply each number of column c(P) by its corresponding r of c(r),
           and put the new products on another column c(R) on the left side 
           of c(P);
       - finally add all numbers of column c(R) to get the final rest R ,
                                                                       n
         while the final quotient will be stated in front of c(k^n)'s 1.
       Therefore:
          A/(k^n) = A  and rest R .
                     n           n

      Remark that any division of an integer number by k^n can be done 
        only by divisions to k, calculations of powers of k, 
        multiplications with 1, 2, ..., k-1, additions.


   Smarandache division is usefull when k is small, the best values being when
   k is an one-digit number, and n large. 
   If k is very big and n verry small, this division becomes useless.


   Example 1 :
     1357/(2^7) = ?


                                         |  /2  |   /2   |
                    ---------------------|------|--------|
                    | c(R) | c(P) | c(r) | c(A) | c(2^7) |
                    |------|------|------|------|--------|
                    |   1  | 2^0  |   1  | 1357 |   2^7  |   line_1
                    |   0  | 2^1  |   0  |  678 |   2^6  |   line_2
                    |   4  | 2^2  |   1  |  339 |   2^5  |   line_3
                    |   8  | 2^3  |   1  |  169 |   2^4  |   line_4
                    |   0  | 2^4  |   0  |   84 |   2^3  |   line_5
                    |   0  | 2^5  |   0  |   42 |   2^2  |   line_6
                    |  64  | 2^6  |   1  |   21 |   2^1  |   line_7
                    |      |      |      --------        |
                    |      |      |      |   10 |   2^0  |   last_line
                    |------|-------------|------|---------
                    |  77  |             


       Therefore:  1357/(2^7) = 10 and rest 77. 
           


      Remark that the division of an integer number by any power of 2 can be 
        done only by divisions to 2, calculations of powers of 2, 
        multiplications and additions.



   Example 2 :
     19495/(3^8) = ?


                                         |  /3   |   /3   |
                    ---------------------|-------|--------|
                    | c(R) | c(P) | c(r) |  c(A) | c(3^8) |
                    |------|------|------|-------|--------|
                    |    1 | 3^0  |   1  | 19495 |   3^8  |   line_1
                    |    0 | 3^1  |   0  |  6498 |   3^7  |   line_2
                    |    0 | 3^2  |   0  |  2166 |   3^6  |   line_3
                    |   54 | 3^3  |   2  |   722 |   3^5  |   line_4
                    |    0 | 3^4  |   0  |   240 |   3^4  |   line_5
                    |  486 | 3^5  |   2  |    80 |   3^3  |   line_6
                    | 1458 | 3^6  |   2  |    26 |   3^2  |   line_7
                    | 4374 | 3^7  |   2  |     8 |   3^1  |   line_8
                    |      |      |      ---------        |
                    |      |      |      |     2 |   3^0  |   last_line
                    |------|-------------|-------|---------
                    | 6373 |             


       Therefore:  19495/(3^8) = 2 and rest 6373. 
           

      Remark that the division of an integer number by any power of 3 can be 
        done only by divisions to 3, calculations of powers of 3, 
        multiplications and additions.


   Reference:
     Alain Bouvier et Michel George, sous la direction de Francois Le Lionnais,
       "Dictionnaire des Mathematiques", Presses Universitaires de France, 
        Paris, 1979, p. 659;
     Colectia "Florentin Smarandache", Arhivele Statului, Filiala Valcea,
        Rm. Valcea, Romania, curator: Ion Soare;
     "The Florentin Smarandache papers" special collection, Arizona State 
        University, Tempe, AZ 85287, USA;
     "The Florentin Smarandache" collection, Texas State University, Center for
        American History, Archives of American Mathematics, Austin, 
        TX 78713, USA.


  112)Let M be a number in a base b.  All distinct digits of M are named
     Smarandache generalized period of M.
    (For example, if M = 104001144, its generalized period is g(M) = {0, 1, 4}.)
    Of course, g(M) is included in {0, 1, 2, ..., b-1}.


  113)The number of Smarandache generalized periods of M is equal to the number
    of the groups of M such that each group contains all distinct digits of M.
    (For example, n (M) = 2 because M = 104 001144.)
                   g                    --- ------
                                         1     2


  114)Length of Smarandache generalized period is equal to the number of its
    distinct digits.
    (For example, l (M) = 3.)
                   g
                              

    Questions:
                                   n  n __
      a) Find n , l  for p  , n!, n , \/n  .
               g   g      n
      b) For a given k >= 1, is there an infinite number of primes p , or n!, or
                                                                    n
          n     n __
         n , or \/n   which have a Smarandache generalized period of length k ?
         Same question such that the number of Smarandache generalized periods  
         be equal to k.
      c) Let a , a , ..., a  be distinct digits.  Is there an infinite number of
              1   2        h
                               n     n __
         primes p , or n!, or n , or \/n  which have as a Smarandache 
                 n
         generalized period the set {a , a , ..., a } ?
                                      1   2        h

   Reference:
     Florentin Smarandache, "Only problems, not solutions!", Xiquan Publishing 
       House, Phoenix, Chicago, 1990, Problem 22, p. 18.



  115)Let {x }    be a sequence of integers, and 0 <= k <= 9 a digit.
            n n>=1
     Smarandache sequence of position is defined as follows:
                        ___                      i
      (k)   (k)         | max {i}, if k is the 10 -th digit of x ;
     U   = U   (x ) =   |                                       n
      n          n      |      -1, otherwise.      
                        ---
     (For example:  if x  = 5, x  = 17, x  = 775, and k = 7, then
                        1       2        3
       (7)   (7)             (7)       (7)
      U   = U   (x ) = -1,  U   = 0,  U   = max {1, 2} = 2.)
       1          1          2         3
                 (k)
      a) Study {U   (p )} , where {p }  is the sequence of primes.  Convergence,
                      n  n          n n
         monotony.
                                                             n
      b) Same question for the sequences:  x  = n! and x  = n  .
                                            n           n

      More generally:  when {x } is a sequence of rational numbers, and k 
                              n n
      belongs to N.
              


  116)Smarandache theorem on characterization of n primes simultaneously:
     Let p  , 1 <= i <= n, 1 <= j <= m , be coprime integers two by two,
          ij                          i
     and let r , ..., r , a , ..., a  be integer numbers such that a  is
              1        n   1        n                               i
     coprime with r  (for each i).
                   i
     For all i one considers the following conditions:
     (i)  p  , ..., p   are simultaneously prime if and only if c  = 0 (mod r ).
           i1        im                                          i           i
                       i
       Then:
     all the numbers p  , 1 <= i <= n, 1 <= j <= m , are simultaneously 
                      ij                          i
     prime if and only if

                 n                                     n
                ___                                   ___
          (R/D) \  (a c /r ) = 0 (mod R/D), where R = | | r , and D is a 
                /    i i  i                           | |  i
                ---                                   i=1
                i=1                                   

     divisor of R.
     
     Corollaries:
       The last relation is equivalent to:

                               m
                 n              i                           
                ___            ___                         
                \   a c  ( P / | | p   ) = 0 (mod P),  
                /    i i       | |  ij              
                ---            j=1
                i=1           

        and
                        
                             m
                 n            i                           
                ___          ___                         
                \   ( a c  / | | p   ) is an integer.  
                /      i i   | |  ij              
                ---          j=1
                i=1           


   Reference:
     Florentin Smarandache, "A general theorem for the characterization of n 
       primes simultaneously", in <Libertas Mathematica>, Texas State 
       University, Arlington, Vol. XI, 1991, pp. 151-5.



  117)Smarandache criterion for twin primes:
     Let p be an odd positive number.  
     Then p and p+2 are twin primes if and only if
                     1      2        1      1
         ( p-1 )! ( --- + ----- ) + --- + -----
                     p    p + 2      p    p + 2
     is an integer.


   Reference:
     Florentin Smarandache, Problem 328, in <College Mathematical Journal>, USA,
       Vol. 17, No. 3, 1986, p. 249.


  118)Smarandache criterion for coprimes:
     If a, b are strictly positive integers, then:
     a and b are coprimes if and only if  

                   F(b)+1     F(a)+1
                  a       +  b       = a + b ( mod ab ),

     where F is Euler's totient.


  119)Smarandache congruence function:
          2
     L : Z  ---> Z, L(x, m) = (x + c ) ... (x + c    ),
                                    1            F(m)
     where F is Euler's totient, and all c , 1 <= i <= F(m), are modulo m
                                          i 
     primitive rest classes.


   Reference:
     Florentin Smarandache, "A numerical function in the congruence theory", in
       <Libertas Mathematica>, Texas State University, Arlington, Vol. XII, 
       1992, pp. 181-5.


  120)Smarandache congruence theorem:
     If a, m are integers, m = 0, then

                        F(m ) + s
                           s         s
                       a          = a   (mod m),

     where F is Euler's totient, and m  and s are obtained by the following
                                      s
     algorithm:

            ---
            | a  = a d  ;   (a , m ) = 1
       (0)  |       0 0       0   0
            |
            | m  = m d  ;    d  = 1
            |       0 0       0
            ---

            ---     1         1
            | d  = d d  ;   (d , m ) = 1
       (1)  |  0    0 1       0   1
            |
            | m  = m d  ;    d  = 1
            |  0    1 1       1
            ---

    ................................................

            ---       1             1
            | d    = d   d    ;   (d   , m   ) = 1
     (s-1)  |  s-2    s-2 s-1       s-2   s-1
            |
            | m    = m   d    ;    d   = 1
            |  s-2    s-1 s-1       s-1
            ---

            ---       1             1
            | d    = d   d    ;   (d   , m ) = 1
       (s)  |  s-1    s-1 s         s-1   s
            |
            | m    = m d      ;    d  = 1.
            |  s-1    s s           s
            ---


     [This is a generalization of Euler's theorem on congruences.]


   Reference:
     Florentin Smarandache, "A generalization of Euler's theorem concerning
       congruences", in <Bulet. Univ. Brasov>, series C, Vol. XXIII, 1982, 
       pp. 37-9;
     Idem, "Une generalisation du theoreme d'Euler", in <Generalisations et
       Generalites>, Ed. Nouvelle, Fes, Morocco, 1984, pp. 9-13.



  121)Smarandache simple functions:
     For any positive prime number p one defines
                                                                        k
        S (k) is the smallest integer such that S (k)! is divisible by p  .
         p                                       p

   Reference:
     Editors of Problem Section, in <Mathematics Magazine>, USA, Vol. 61, No. 3,
       June 1988, p. 202.


  122)Smarandache function:
     S(n) is defined as the smallest integer such that S(n)! is divisible by n
     (for n = 0).

     If the canonical factorization of n is

                k     k
                 1     s
           n = p ... p
                1     s


     then S(n) = max { S (k ) }, where S   are Smarandache simple functions.
                        p  i            p
                         i               i


   References:
     M. Andrei, I. Balacenoiu, V. Boju, E. Burton, C. Dumitrescu, Jim Duncan, 
       Pal Gronas, Henry Ibstedt, John McCarthy,  Mike Mudge, Marcela Popescu, 
       Paul Popescu, E. Radescu, N. Radescu, V. Seleacu, J. R. Sutton, 
       L. Tutescu, Nina Varlan, St. Zanfir, and others,
       in <Smarandache Function Journal>, Department of Mathematics, University
       of Craiova, 1993-4.


  123)Smarandache prime equation conjecture:
     For any k >= 2, the diophantine equation

               y = 2X X ... X  + 1
                     1 2     k

     has an infinite number of prime solutions.

     (For example:  when k = 3, 647 = 2x17x19+1;
                    when k = 4, 571 = 2x3x5x19+1, etc.)


   Reference:
     Florentin Smarandache, "Only problems, not solutions!", Xiquan Publishing 
       House, Phoenix, Chicago, 1990, Problem 11.



  124)Smarandache progressions:
     How many primes do the following progressions contain:
       a) { ap + b }, n = 1, 2, 3, ..., where (a, b) = 1 and p  is the n-th
              n                                               n

          prime?

             n
       b) { a  + b }, n = 1, 2, 3, ..., where (a, b) = 1, and a is different  

          from -1, 0, +1 ?

             n              n
       c) { n  + 1 } and { n  - 1 }, n = 1, 2, 3, ... ?


   Reference:
     Florentin Smarandache, "Only problems, not solutions!", Xiquan Publishing
       House, Phoenix, Chicago, 1990, Problem 17.


  125)Smarandache inequality:

                   k-1 __   __
             n-k+1 ___ | n-i |
       n! > k      | | | --- |!
                   i=0 |  k  |
                       --   --

     for any non-null positive integers n and k.

     If k =2 (for example), one obtains:
      
                  __   __  __   __
             n-1  | n-1 |  |  n  |   
       n! > 2     | --- |! | --- |!
                  |  2  |  |  2  |
                  --   --  --   --

     and if k = 3 (another example), one obtains:

                  __   __  __   __  __   __
             n-2  | n-2 |  | n-1 |  |  n  |
       n! > 3     | --- |! | --- |! | --- |!
                  |  3  |  |  3  |  |  3  |
                  --   --  --   --  --   --


   Reference:   
     Florentin Smarandache, "Problemes avec et sans ... problemes!", Somipress,
       Fes, Morocco, 1983, Problemes 7.88 & 7.89, pp. 110-1.



  126)Smarandache divisibility Theorem:
     If a and m are integers, and m > 0, then 

                 m
              ( a  - a ) ( m - 1 )!

     is divisible by m.


   Reference:
     Florentin Smarandache, "Problemes avec et sans ... problemes!", Somipress,
       Fes, Morocco, 1983, Problemes 7.140, pp. 173-4.


 
  127)Smarandache dilemmas:
       Is it true that for any question there exists at least an answer?
     Reciprocally:
       Is any assertion the result of at least a question?


   Reference:
     Florentin Smarandache, "Only problems, not solutions!", Xiquan Publishing
       House, Phoenix, Chicago, 1990, Problem 5, p. 8.



  128)Smarandache surface points:
     a) Let n be an integer >= 5.  Find a minimum number M(n) such that anyhow
        are choosen M(n) points in space, four by four noncoplanar, there exist 
        n points among these which belong to the surface of a sphere.
     b) Same question for an arbitrary space body (for example: cone, cube, 
        etc.).
     c) Same question in plane (for n >= 4, and the points are choosen three 
        by three noncolinear).


   Reference:
     Florentin Smarandache, "Only problems, not solutions!", Xiquan Publishing
       House, Phoenix, Chicago, 1990, Problem 9, p. 10.


  129)Smarandache inclusion problems:
     a) Find a method to get the maximum number of circles of radius 1 included
        in a given planar figure, at most tangential two by two or tangential
        to the border of the planar figure.
        Study the general problem when "circle" are replaced by an arbitrary
        planar figure.
     b) Same question for spheres of radius 1 included in a given space body.
        Study the general problem when "sphere" are replaced by an arbitrary
        space body.


   Reference:
     Florentin Smarandache, "Only problems, not solutions!", Xiquan Publishing
       House, Phoenix, Chicago, 1990, Problem 8, p. 10.



  130)Smarandache convex polyhedrons:
     a) Given n points in space, four by four noncoplanar, find the maximum 
        number M(n) of points which constitute the vertexes of a convex 
        polyhedron.
          Of course, M(n) >= 4.
     b) Given n points in space, four by four noncoplanar, find the minimum
        number N(n) >= 5 such that: any N(n) points among these do not 
        constitute the vertexes of a convex polyhedron.
          Of course, N(n) may not exist.


   Reference:
     Florentin Smarandache, "Only problems, not solutions!", Xiquan Publishing
       House, Phoenix, Chicago, 1990, Problem 7, p. 9;
     Ioan Tomescu, "Problems of combinatorics and grsph Theory" (Romanian), 
       Bucharest, Editura Didactica si Pedagogica, 1983.



  131)Smarandache integral points:
     How many noncoplanar points in space can be drawn at integral distances 
     each from other?
     Is it possible to find an infinite number of such points?



  132)Smarandache counter:
     C(a, b) = how many digits of "a" the number b contains.

     Study, for example, C(1, p ), where p  is the n-th prime.
                               n          n

                                n
     Same for:  C(1, n!), C(1, n ).



   Reference:
     Florentin Smarandache, "Only problems, not solutions!", Xiquan Publishing
       House, Phoenix, Chicago, 1990, Problem 3, p. 7.



  133)Smarandache maximum points:
     Let d > 0.  Question:
     What is the maximum number of points included in a given planar figure
     (generally: in a space body) such that the distance between any two 
     points is greater or equal than d ?


  134)Smarandache minimum points:
     Let d > 0.  Question:
     What is the minimum number of points {A , A , ... }  included in a given 
                                            1   2  
     planar figure (generally: in a space body) such that if another point A 
     is included in that figure then there exists at least an A  with the 
                                                               i  
     distance |AA | < d ?
                 i


   Reference:
     Florentin Smarandache, "Only problems, not solutions!", Xiquan Publishing
       House, Phoenix, Chicago, 1990, Problem 2, p. 6.



  135)Smarandache increasing repeated compositions:
     Let g be a function, g : N ---> N, such that g(n) > n for all natural n.
     An increasing repeated composition related to g and a given positive
     number m is defined as below:

        F  : N ---> N, F (n) = k, where k is the smallest integer such that
         g              g   
 
        g(...g(n)...) >= m (g is composed k times).


     Study, for example, F , where s is the function that associates to each
                          s
     non-null positive integer n the sum of its positive divisors.


  136)Smarandache decreasing repeated compositions:
     Let g be a non-constant function, g : N ---> N, such that g(n) <= n for 
     all natural n.
     An decreasing repeated composition related to g is defined as below:

        f  : N ---> N, f (n) = k, where k is the smallest integer such that
         g              g   
 
        g(...g(n)...) = constant (g is composed k times).


     Study, for example, f , where d is the function that associates to each
                          d
     non-null positive integer n the number of its positive divisors.
     In this particular case, the constant is 2.
              ___
     Same for | |(n) = the number of primes not exceeding n,
     and for p(n) = the largest prime factor of n,
     and for o(n) = the number of distinct prime factors of n.


   Reference:
     Florentin Smarandache, "Only problems, not solutions!", Xiquan Publishing
       House, Phoenix, Chicago, 1990, Problems 18, 19 & 29, pp. 15-6, 23.



  137)Smarandache coloration conjecture:
     Anyhow all points of an m-dimensional euclidian space are colored with a
     finite number of colors, there exists a color which fulfills all distances.


   Reference:
     Florentin Smarandache, "Only problems, not solutions!", Xiquan Publishing
       House, Phoenix, Chicago, 1990, Problem 13, p. 12.



  138)Smarandache primes:
     Let a , ..., a  be distinct digits, 1 <= n <= 9.
          1        n
     How many primes can we construct from all these digits only (eventually
     repeated) ?
     
     (More generally: when a , ..., a , and n are positive integers.)
                            1        n


   Reference:
     Florentin Smarandache, "Only problems, not solutions!", Xiquan Publishing
       House, Phoenix, Chicago, 1990, Problem 3, p. 7.



  139)Smarandache prime numbers theorem:
     There exist an infinite number of primes which contain given digits,
     a , a , ..., a , in the positions i , i , ..., i ,
      1   2        m                    1   2        m
                                                                i
     with i  > i  > ... > i  >= 0 (the "i-th position" is the 10  -th digit).
           1    2          m
     (Of course, if i = 0, then a  must be odd and different from 5.)
                     m           m


   Reference:
     Florentin Smarandache, Query 762 (and Answer 762), in <Mathematics
       Magazine>, April, 1990.
     Florentin Smarandache, "Only problems, not solutions!", Xiquan Publishing
       House, Phoenix, Chicago, 1990, Problem 10, p. 11.



  140)Smarandache cardinality Theorem:
     For any positive integers n >= 1 and m >= 3, find the maximum number 
     S(n ,m) such that:
     the set { 1, 2, 3, ..., n} has a subset A of cardinality S(n ,m) with
     the property that A contains no m-term arithmetic progression.
       S(n ,m) is called the Smarandache cardinality number.
    
     Study it.



   References:
     Florentin Smarandache, "Arithmetic progressions: Problem 88-5", in
       <The Mathematical Intelligencer>, Vol. 11, No. 1, 1989;
     E. Kurt Tekolste (Wayne, PA), "Solution to Problem 88-5", in
       <The Mathematical Intelligencer>, Vol. 11, No. 1, 1989.


 
            Contents:


  1)Smarandache Consecutive Sequence
  2)Smarandache Circular Sequence
  3)Smarandache Symmetric Sequence
  4)Smarandache Deconstructive Sequence 
  5)Smarandache Mirror Sequence
  6)Smarandache Permutation Sequence
  7)Smarandache Generalized Permutation Sequence
  8)Smarandache Mobile Periodicals (I)
  9)Smarandache Mobile Periodicals (II)
  10)Smarandache Infinite Numbers (I)
  11)Smarandache Infinite Numbers (II) 
  12)Smarandache Car
  13)Smarandache Finite Lattice
  14)Smarandache Infinite Lattice
  15)Smarandache Simple Numbers
  16)Smarandache Digital Sum
  17)Smarandache Digital Products
  18)Smarandache Code Puzzle
  19)Smarandache Pierced Chain
  20)Smarandache Divisor Products
  21)Smarandache Proper Divisor Products
  22)Smarandache Square Complements
  23)Smarandache Cubic Complements
  24)Smarandache m-Power Complements
  25)Smarandache Cube Free Sieve
  26)Smarandache m-Power Free Sieve
  27)Smarandache Irrational Root Sieve
  28)Smarandache Odd Sieve
  29)Smarandache Binary Sieve
  30)Smarandache Trinary Sieve
  31)Smarandache n-ary Sieve  (generalization, n >= 2)
  32)Smarandache Consecutive Sieve
  33)Smarandache General-Sequence Sieve
  34)Smarandache More General-Sequence Sieve
  35)Smarandache Sequences of Sequences: 
      Digital Sequences & Construction Sequences
  36)Smarandache General Residual Sequence
  37)Smarandache (Inferior) Prime Part
  38)Smarandache (Superior) Prime Part
  39)Smarandache (Inferior) Square Part
  40)Smarandache (Superior) Square Part
  41)Smarandache (Inferior) Cube Part
  42)Smarandache (Superior) Cube Part
  43)Smarandache (Inferior) Factorial Part
  44)Smarandache (Superior) Factorial Part
  45)Smarandache Double Factorial Complements
  46)Smarandache Prime Additive Complements
  47)Smarandache Prime Base
  48)Smarandache Square Base
  49)Smarandache m-Power Base  (generalization)
  50)Smarandache Factorial Base
  51)Smarandache Generalized Base
  52)Smarandache Numbers
  53)Smarandache Quotients
  54)Smarandache Double Factorial Numbers
  55)Smarandache Primitive Numbers (of Power 2)
  56)Smarandache Primitive Numbers (of Power 3)
  57)Smarandache Primitive Numbers (of Power p, p prime)  {generalization}
  58)Smarandache Square Residues
  59)Smarandache Cubical Residues
  60)Smarandache m-Power Residues   {generalization}
  61)Smarandache Exponents (of Power 2)
  62)Smarandache Exponents (of Power 3)
  63)Smarandache Exponents (of Power p)  {generalization}
  64)Smarandache Pseudo-Primes of First Kind
  65)Smarandache Pseudo-Primes of Second Kind
  66)Smarandache Pseudo-Primes of Third Kind
  67)Smarandache Almost Primes of First Kind
  68)Smarandache Almost Primes of Second Kind
  69)Smarandache Pseudo-Squares of First Kind
  70)Smarandache Pseudo-Squares of Second Kind
  71)Smarandache Pseudo-Squares of Third Kind
  72)Smarandache Pseudo-Cubes of First Kind
  73)Smarandache Pseudo-Cubes of Second Kind
  74)Smarandache Pseudo-Cubes of Third Kind
  75)Smarandache Pseudo-m-Powers of First Kind
  76)Smarandache Pseudo-m-Powers of Second Kind
  77)Smarandache Pseudo-m-Powers of Third Kind
  78)Smarandache Pseudo-Factorials of First Kind
  79)Smarandache Pseudo-Factorials of Second Kind
  80)Smarandache Pseudo-Factorials of Third Kind
  81)Smarandache Pseudo-Divisors of First Kind
  82)Smarandache Pseudo-Divisors of Second Kind
  83)Smarandache Pseudo-Divisors of Third Kind
  84)Smarandache Pseudo-Odd Numbers of First Kind
  85)Smarandache Pseudo-Odd Numbers of Second Kind
  86)Smarandache Pseudo-Odd Numbers of Third Kind
  87)Smarandache Pseudo-Triangular Numbers
  88)Smarandache Pseudo-Even Numbers of First Kind
  89)Smarandache Pseudo-Even Numbers of Second Kind
  90)Smarandache Pseudo-Even Numbers of Third Kind
  91)Smarandache Pseudo-Multiples of First Kind (of 5)
  92)Smarandache Pseudo-Multiples of Second Kind (of 5)
  93)Smarandache Pseudo-Multiples of Third Kind (of 5)
  94)Smarandache Pseudo-Multiples of First Kind of p (p >= 2)
  95)Smarandache Pseudo-Multiples of Second Kind of p (p >= 2)
  96)Smarandache Pseudo-Multiples of Third Kind of p (p >= 2)
  97)Smarandache Constructive Set (of Digits 1,2)
  98)Smarandache Constructive Set (of Digits 1,2,3)
  99)Smarandache Generalized Constructive Set
  100)Smarandache Square Roots
  101)Smarandache Cubical Roots
  102)Smarandache m-Power Roots
  103)Smarandache Numerical Carpet
  104)Goldbach-Smarandache Table
  105)Smarandache-Vinogradov Table
  106)Smarandache-Vinogradov Sequence
  107)Smarandache Paradoxist Numbers
  108)Non-Smarandache Numbers
  109)The Paradox of Smarandache Numbers
  110)Smarandache Multiplication
  111)Smarandache Division by k^n (k, n >= 2)
  112)Smarandache Generalized Period of a Number
  113)The Number of Smarandache Generalized Periods
  114)Length of Smarandache Generalized Period
  115)Smarandache Sequence of Position
  116)Smarandache Theorem on Characterization of n Primes Simultaneously
  117)Smarandache Criterion for Twin Primes
  118)Smarandache Criterion for Coprimes
  119)Smarandache Congruence Function
  120)Smarandache Congruence Theorem
  121)Smarandache Simple Functions
  122)Smarandache Function
  123)Smarandache Prime Equation Conjecture
  124)Smarandache Progressions
  125)Smarandache Inequality
  126)Smarandache Divisibility Theorem
  127)Smarandache Dilemmas
  128)Smarandache Surface Points
  129)Smarandache Inclusion Problems
  130)Smarandache Convex Polyhedrons
  131)Smarandache Integral Points
  132)Smarandache Counter
  133)Smarandache Maximum Points
  134)Smarandache Minimum Points
  135)Smarandache Increasing Repeated Compositions
  136)Smarandache Decreasing Repeated Compositions
  137)Smarandache Coloration Conjecture
  138)Smarandache Primes
  139)Smarandache Prime Numbers Theorem
  140)Smarandache Cardinality Theorem