(as received from Steve Finch), 



Magata's constant

3.4070691656272561422194582628271806535540344380150321161910338275729699387


Starting with the dataset

(1,2), (2,3), (3,5), (4,7), (5,11), (6,13), ..., (n,p_n)

where p_n is the nth prime number, Frederick Magata (1998) used Newtonian
interpolation 
to determine the coefficients b_k of a (n-1) degree polynomial fit

b_0 + b_1*(x-1) + b_2*(x-1)*(x-2) + b_3*(x-1)*(x-2)*(x-3) + ...

The sum of all the coefficients b_k, for arbitrarily large n, appears to
converge
to 3.407069...  Here is a Mathematica program which verifies this claim,
along 
with the values of b_k, 0<=k<=9.


n:=100

Prms:=Array[Prime,n]

P[y_]:=P[y]=InterpolatingPolynomial[Prms,x]/.x->y

N[Sum[Level[P[y],{2*k-1}][[1]],{k,1,n-1}]+Level[P[y],{2*n-2}][[1]],75]
3.4070691656272561422194582628271806535540344380150321161910338275729699387

Do[Print[Level[P[y],{2*k-1}][[1]]], {k,1,10}]
2
1
1
-
2
  1
-(-)
  6
1
-
8
  3
-(--)
  40
23
---
720
   53
-(----)
  5040
 23
----
8064
    79
-(------)
  120960






						                 


































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