Comments from Randall L. Rathbun (randallr(AT)nethere.com), Jan 19 2002 on A058047:
%I A058047
%S A058047 3,5,7,29,41,79,
%N A058047 Class 1 primes of the Murad Sequence
%C A058047 The Murad Sequence is a generalized Collatz Sequence:
For a predefined prime p and a number a_n:
let a_(n+1) = a_n/P(q_i), for all primes q_i<p such that q_i divides a_n
or a_(n+1) = p.a_n + 1, if no one prime q_i<p divides a_n
continue until a(x) = 1 or a cycle lockup occurs.
Example:
if p = 5 and a0 = 35, the Murad's sequence is: {35,176,88,44,22,11,56,28,14,7,36,6,1}
i.e.
35 is not divisible by 3 or 2, so we multiply by 5 and add +1
176 is divisible by just 2 so we reduce to 88
88 is divisible by 2, so we reduce to 44
44, 22, 11 similarly
11 is not divisible by either 3,2 so we multiply by 5 and add +1 = 11*5+1 = 56
56, 28, 14, are all divisible by 2.. we reduce 14 by 2 to get 7
7 is not divisible by either 3,2 so we get 5*7+1 = 36
36 is divisible by both 3 and 2 so we have 36/3 = 12/2 = 6
6 is divisible again by both 3,2 so we have 6/3 = 2/2 = 1
1 is the terminating number so we're done
As you can easily recognize from the Murad's generalization formulas, the Murad's sequences for p = 3 are the same as the Collatz's sequences.
Murad has found the following conjectural empirical facts:
1.- The odd primes can be divided in two classes:
a) Class I: the primes such that the sequences linked to them has a behavior similar to p=3, that is to say for any a0 starting value the sequence arrives to "1". This class of primes is composed for p = 3, 5, 7, 19,...
b) Class II: the primes such that the sequences linked to them for certain values a0 does not arrive to "1" but get 'hooked' to certain "cycles". This class of primes is composed for the primes p = 11, 13, 17,...
NOTE: Jeff Heleen showed that 19 is NOT in the class 1 primes.
See the web page http://www.primepuzzles.net/puzzles/puzz_114.htm