A191997 Denominators of partial products of a Hardy-Littlewood constant.

1, 2, 32, 128, 512, 8192, 2097152, 226492416, 301989888, 536870912, 32212254720, 8349416423424, 4453022092492800, 1122161567308185600, 2294196982052290560, 12235717237612216320, 16314289650149621760, 58731442740538638336000, 51166832915557261718323200

Offset

2,2

Comments

The rational partial products are r(n)=A191996(n)/a(n), n>=1.

The limit r(n), n->infinity, approximately 1.3203236, is the constant C(f_1,f_2) appearing in the Hardy-Littlewood conjecture (also called Bateman-Horn conjecture) for the integer polynomials f_1=x and f_2=x+2 (relevant for twin primes). See the Conrad reference Example 1, p. 134, also for the original references.

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=2..20.

Wolfdieter Lang, Rationals and limit.

Formula

a(n) = denominator(r(n)), with the rational r(n):=2*product(1-1/(p(j)-1)^2,j=2..n), with the primes p(j):=A000040(j).

Example

The rationals r(n)(in lowest terms) are 2, 3/2, 45/32, 175/128, 693/512, 11011/8192,...

Crossrefs

A191996, A191998/A191999

Sequence in context: A303078 A226395 A257965 * A274654 A112850 A123105

Adjacent sequences: A191994 A191995 A191996 * A191998 A191999 A192000

Keyword

nonn,easy,frac

Author

Wolfdieter Lang, Jun 21 2011

Status

approved