A191998 Numerators of partial products of a Hardy-Littlewood constant.

1, 3, 9, 21, 231, 847, 2541, 16093, 33649, 43263, 447051, 1043119, 13560547, 83300503, 170222767, 222599003, 13133341177, 774867129443, 4719645242971, 335094812250941

Offset: 1

Author

Wolfdieter Lang, Jun 21 2011

Keywords

• nonn
• easy
• frac

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).

Examples

			The rationals r(n) are: 1, 3/2, 9/8, 21/16, 231/160, 847/640, 2541/2048, ...
		

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.

Links

• Wolfdieter Lang, Rationals and limit.

Crossrefs

Cf. A191999, A191996/A191997.

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).