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A191998
Numerators of partial products of a Hardy-Littlewood constant.
4
1, 3, 9, 21, 231, 847, 2541, 16093, 33649, 43263, 447051, 1043119, 13560547, 83300503, 170222767, 222599003, 13133341177, 774867129443, 4719645242971, 335094812250941
OFFSET
1,2
COMMENTS
The rational partial products are r(n) = a(n)/A191999(n), n >= 1.
The limit r(n), n->infinity, approximately 1.3728134 is the constant C(f) appearing in the Hardy-Littlewood conjecture (also called Bateman-Horn conjecture) for the integer polynomial f=x^2+1. See the Conrad reference Example 2, p. 134, also for the original references.
Note that the nontrivial Dirichlet character modulo 4, called Chi_2(4;n) = A056594(n-1), n >= 1, appears as Chi_4(n) in this reference. The constant 0.6864067 given there is C(f)/2 (the degree of the function f has been divided).
REFERENCES
Keith Conrad, Hardy-Littlewood constants, pp. 133-154 in: Mathematical properties of sequences and other combinatorial structures, edts. Jong-Seon No et al., Kluwer, Boston/Dordrecht/London, 2003.
FORMULA
a(n) = numerator(r(n)) with
r(n) := product(1-Chi_2(4;p(j))/(p(j)-1),j=1..n), n>=1, with the primes p(j)=A000040(j) and the nontrivial Dirichlet Character modulo 4, called here Chi_2(4;k) = A056594(k).
EXAMPLE
The rationals r(n) are: 1, 3/2, 9/8, 21/16, 231/160, 847/640, 2541/2048, ...
CROSSREFS
KEYWORD
nonn,easy,frac
AUTHOR
Wolfdieter Lang, Jun 21 2011
STATUS
approved