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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 (list; graph; refs; listen; history; text; internal format)
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 .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.

LINKS

Table of n, a(n) for n=1..20.

Wolfdieter Lang, Rationals and limit.

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

Cf. A191999, A191996/A191997.

Sequence in context: A252284 A259367 A193374 * A098980 A007647 A247182

Adjacent sequences:  A191995 A191996 A191997 * A191999 A192000 A192001

KEYWORD

nonn,easy,frac

AUTHOR

Wolfdieter Lang, Jun 21 2011

STATUS

approved

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Last modified January 16 06:59 EST 2019. Contains 319188 sequences. (Running on oeis4.)