A154945 Decimal expansion of Sum_{p} 1/(p^2-1), summed over the primes p = A000040.

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

Offset

0,1

Comments

By geometric series expansion, the same as the sum over the prime zeta function at even arguments, P(2i), i=1,2,....

(Pi^2/6)*density of A190641, the numbers divisible by exactly one prime with exponent greater than 1. - Charles R Greathouse IV, Aug 02 2016

Links

Table of n, a(n) for n=0..104.

Jacques Grah, Comportement moyen du cardinal de certains ensembles de facteurs premiers, Monatsh. Math., Vol. 118 (1994), pp. 91-109, Corollary 6.1.

Carl Pomerance and Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory, Vol. 1, Iss. 1 (2011), pp. 52-66. See p. 61.

Index to constants which are prime zeta sums {0,1,1}

Formula

Equals Sum_{k>=1} 1/A084920(k) = Sum_{i>=1} P(2i) = A085548+A085964+A085966+A085968+... = A152447+A085548-A154932.

Equals Sum_{k>=2} 1/A000961(k)^2 = Sum_{k>=2} 1/A056798(k). - Amiram Eldar, Sep 21 2020

Equals (A136141 + A179119)/2. - Artur Jasinski, Mar 31 2025

Equals lim_{m->oo} (1/m^2) * Sum_{i,j=1..m} Omega(gcd(i, j)), where Omega = A001222. - Amiram Eldar, Mar 14 2026

Example

0.551693297656999184439731023971343578131500377778628252230...

Mathematica

digits = 105; m0 = 2 digits; Clear[rd]; rd[m_] := rd[m] = RealDigits[delta1 = Sum[PrimeZetaP[2n], {n, 1, m}] , 10, digits][[1]]; rd[m0]; rd[m = 2m0];
While[rd[m] != rd[m-m0], Print[m]; m = m+m0]; Print[N[delta1, digits]]; rd[m] (* Jean-François Alcover, Sep 11 2015, updated Mar 16 2019 *)

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))))
sumpos(n=1, primezeta(2*n)) \\ Charles R Greathouse IV, Aug 02 2016
(PARI) sumeulerrat(1/(p^2-1)) \\ Amiram Eldar, Mar 18 2021

Crossrefs

Cf. A000040, A000961, A001222, A056798, A084920, A085548, A085964, A085966, A085968, A152447, A154932, A190641.

Sequence in context: A190101 A097566 A387206 * A300710 A254347 A011094

Adjacent sequences: A154942 A154943 A154944 * A154946 A154947 A154948

Keyword

cons,nonn

Author

R. J. Mathar, Jan 17 2009

Extensions

More digits from Jean-François Alcover, Sep 11 2015

Status

approved