%I #6 Mar 30 2012 18:49:34
%S 1,2,32,128,512,8192,2097152,226492416,301989888,536870912,
%T 32212254720,8349416423424,4453022092492800,1122161567308185600,
%U 2294196982052290560,12235717237612216320,16314289650149621760,58731442740538638336000,51166832915557261718323200
%N Denominators of partial products of a Hardy-Littlewood constant.
%C The rational partial products are r(n)=A191996(n)/a(n), n>=1.
%C 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.
%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="/A191997/a191997.txt">Rationals and limit.</a>
%F 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).
%e The rationals r(n)(in lowest terms) are 2, 3/2, 45/32, 175/128, 693/512, 11011/8192,...
%Y A191996, A191998/A191999
%K nonn,easy,frac
%O 2,2
%A _Wolfdieter Lang_, Jun 21 2011