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A271971 Decimal expansion of (6/Pi^2) Sum_{p prime} 1/(p(p+1)), a Meissel-Mertens constant related to the asymptotic density of certain sequences of integers.

%I A271971 #12 Mar 18 2021 07:34:11
%S A271971 2,0,0,7,5,5,7,2,2,0,1,9,2,6,5,9,8,6,9,9,6,2,5,0,7,2,3,1,1,4,4,0,4,7,
%T A271971 6,5,8,5,3,5,3,5,5,5,5,3,5,2,5,6,1,9,1,6,1,5,9,7,6,3,2,9,8,3,6,5,2,5,
%U A271971 4,0,7,4,7,4,7,9,6,4,9,7,9,1,2,1,1,9,0,9,4,2,6,8,4,5,0,3,5,9,4,6
%N A271971 Decimal expansion of (6/Pi^2) Sum_{p prime} 1/(p(p+1)), a Meissel-Mertens constant related to the asymptotic density of certain sequences of integers.
%C A271971 This is the density of A060687, the numbers with one 2 and the rest 1s in the exponents of its prime factorization. - _Charles R Greathouse IV_, Aug 03 2016
%D A271971 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.2 Meissel-Mertens Constants, p. 95.
%F A271971 Equals (6/Pi^2)*A179119.
%e A271971 0.200755722019265986996250723114404765853535555352561916...
%t A271971 digits = 100; S = (6/Pi^2)*NSum[(-1)^n PrimeZetaP[n], {n, 2, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits+5]; RealDigits[ S, 10, digits] // First
%o A271971 (PARI) eps()=2.>>bitprecision(1.)
%o A271971 primezeta(s)=my(t=s*log(2)); sum(k=1, lambertw(t/eps())\t, moebius(k)/k*log(abs(zeta(k*s))))
%o A271971 sumalt(k=2, (-1)^k*primezeta(k))*6/Pi^2 \\ _Charles R Greathouse IV_, Aug 03 2016
%o A271971 (PARI) sumeulerrat(1/(p*(p+1)))/zeta(2) \\ _Amiram Eldar_, Mar 18 2021
%Y A271971 Cf. A059956, A060687, A085548, A136141, A179119, A222056.
%K A271971 nonn,cons
%O A271971 0,1
%A A271971 _Jean-François Alcover_, Apr 17 2016