A246061 Decimal expansion of lim_{n->infinity} ((1/log(n)^2)*Product_{2 < p < n, p prime} p/(p-2)).

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, 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, 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

Offset: 1

Author

Jean-François Alcover, Sep 11 2014

Keywords

• nonn
• cons

Examples

			1.201303559967362241247555959207383482453838449427113...
		

References

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

Links

• Eric Weisstein's MathWorld, Twin Primes Constant.

Crossrefs

Cf. A005597, A091724.

Programs

Mathematica

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

PARI

exp(2*Euler)/(4*prodeulerrat(1-1/(p-1)^2, 1, 3)) \\ Amiram Eldar, Apr 27 2025

Formula

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