A154932 -id:A154932 - cp's OEIS frontend

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

R. J. Mathar, Jan 17 2009

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

			0.551693297656999184439731023971343578131500377778628252230...

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}

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

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 *)

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

sumeulerrat(1/(p^2-1)) \\ Amiram Eldar, Mar 18 2021

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

More digits from Jean-François Alcover, Sep 11 2015

A284748 Decimal expansion of the sum of reciprocals of composite powers.

Original entry on oeis.org

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

Offset: 0

Terry D. Grant, Apr 01 2017

			Equals 1/(4*3)+1/(6*5)+1/(8*7)+1/(9*8)+1/(10*9)+...
= 0.226843330950204872135632540144057604...

Cf. A066247, A077761, A179119, A185380.
Decimal expansion of the sum of reciprocal powers: A136141 (primes), A154945 (primes at even powers), A152447 (semiprimes), A154932 (squarefree semiprimes).
Decimal expansion of the 'nonprime zeta function': A275647 (at 2), A278419 (at 3).

RealDigits[ NSum[Zeta[n]-1-PrimeZetaP[n], {n, 2, Infinity}], 10, 105] [[1]]

1 - sumeulerrat(1/(p*(p-1))) \\ Amiram Eldar, Mar 18 2021

Equals Sum_{n>=1} 1/A002808(n)^(n+1) = (A275647 - 1) + (A278419 - 1) + ...
Equals Sum_{n>=1} 1/A002808(n)*(A002808(n)-1).
Equals Sum_{n>=2} (Zeta(n) - PrimeZeta(n) - 1) = Sum_{n>=2} CompositeZeta(n).
Equals 1 - A136141.

More digits from Vaclav Kotesovec, Jan 13 2021

cp's OEIS Frontend

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

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A284748 Decimal expansion of the sum of reciprocals of composite powers.

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