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.

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, 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, 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

Offset

0,1

Comments

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

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.2 Meissel-Mertens Constants, p. 95.

Links

Table of n, a(n) for n=0..99.

Formula

Equals (6/Pi^2)*A179119.

Example

0.200755722019265986996250723114404765853535555352561916...

Mathematica

digits = 100; S = (6/Pi^2)*NSum[(-1)^n PrimeZetaP[n], {n, 2, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits+5]; RealDigits[ S, 10, digits] // First

Prog

(PARI) eps()=2.>>bitprecision(1.)
primezeta(s)=my(t=s*log(2)); sum(k=1, lambertw(t/eps())\t, moebius(k)/k*log(abs(zeta(k*s))))
sumalt(k=2, (-1)^k*primezeta(k))*6/Pi^2 \\ Charles R Greathouse IV, Aug 03 2016
(PARI) sumeulerrat(1/(p*(p+1)))/zeta(2) \\ Amiram Eldar, Mar 18 2021

Crossrefs

Cf. A059956, A060687, A085548, A136141, A179119, A222056.

Sequence in context: A370796 A094596 A143024 * A278157 A198232 A160213

Adjacent sequences: A271968 A271969 A271970 * A271972 A271973 A271974

Keyword

nonn,cons

Author

Jean-François Alcover, Apr 17 2016

Status

approved