A246061 - OEIS

%I #33 Dec 20 2017 04:10:51

%S 1,2,0,1,3,0,3,5,5,9,9,6,7,3,6,2,2,4,1,2,4,7,5,5,5,9,5,9,2,0,7,3,8,3,

%T 4,8,2,4,5,3,8,3,8,4,4,9,4,2,7,1,1,3,0,8,5,1,8,1,9,5,5,9,7,4,1,4,8,0,

%U 0,9,9,7,7,9,4,3,7,7,5,2,2,5,9,6,7,0,6,4,3,1,8,4,8,6,1,9,7,6,0,8,8

%N Decimal expansion of lim_{n->infinity} ((1/log(n)^2)*Product_{2 < p < n, p prime} p/(p-2)).

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.1 Hardy-Littlewood constants, p. 86.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/TwinPrimesConstant.html">Twin Primes Constant</a>

%F Equals exp(2*EulerGamma)/(4*C_2), where C_2 is the twin primes constant A005597.

%e 1.201303559967362241247555959207383482453838449427113...

%t digits = 101; s[n_] := (1/n)* N[Sum[MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], digits + 60]; C2 = (175/256)*Product[(Zeta[ n]*(1 - 2^(-n))*(1 - 3^(-n))*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[ n]), {n, 2, digits + 60}]; RealDigits[Exp[2*EulerGamma]/(4*C2), 10, digits] // First

%Y Cf. A005597.

%K nonn,cons

%O 1,2

%A _Jean-François Alcover_, Sep 11 2014