A191996 Numerators of partial products of a Hardy-Littlewood constant.

2, 3, 45, 175, 693, 11011, 2807805, 302307005, 402243205, 714186915, 42803602439, 11086133031701, 5908908905896633, 1488200914442251997, 3041106216468949733, 16213234917387714257, 21611220383343195817, 77778782159652161745383, 67745319261057032880228593

Offset: 2

Author

Wolfdieter Lang, Jun 21 2011

Keywords

• nonn
• easy
• frac

Comments

The rational partial products are r(n)=a(n)/A191997(n), n>=1.

The limit r(n), n->infinity, approximately 1.3203236 = A114907, 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.

Essentially the same as A062270. - R. J. Mathar, Jun 23 2011

Examples

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

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

A191997, A191998/A191999.

Formula

a(n) = numerator(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).