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%I #14 Feb 11 2020 02:06:11
%S 1,3,9,21,231,847,2541,16093,33649,43263,447051,1043119,13560547,
%T 83300503,170222767,222599003,13133341177,774867129443,4719645242971,
%U 335094812250941
%N Numerators of partial products of a Hardy-Littlewood constant.
%C The rational partial products are r(n) = a(n)/A191999(n), n >= 1.
%C 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.
%C 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).
%D 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.
%H Wolfdieter Lang, <a href="/A191998/a191998.txt">Rationals and limit.</a>
%F a(n) = numerator(r(n)) with
%F 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).
%e The rationals r(n) are: 1, 3/2, 9/8, 21/16, 231/160, 847/640, 2541/2048, ...
%Y Cf. A191999, A191996/A191997.
%K nonn,easy,frac
%O 1,2
%A _Wolfdieter Lang_, Jun 21 2011
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