A343613 Decimal expansion of P_{3,2}(3) = Sum 1/p^3 over primes == 2 (mod 3).

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

Offset

0,2

Comments

The prime zeta modulo function P_{m,r}(s) = Sum_{primes p == r (mod m)} 1/p^s generalizes the prime zeta function P(s) = Sum_{primes p} 1/p^s.

Links

Jean-François Alcover, Table of n, a(n) for n = 0..1006

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=3, n=2, s=3) on p. 21.

OEIS index to entries related to the (prime) zeta function.

Formula

P_{3,2}(3) = P(3) - 1/3^3 - P_{3,1}(3) = A085541 - A021031 - A175645.

Example

0.134125178915463540428599329999431198995879919752168337370599106153853349956...

Prog

(PARI) s=0; forprimestep(p=2, 1e8, 3, s+=1./p^3); s \\ For illustration: using primes up to 10^N gives about 2N+2 (= 18 for N=8) correct digits.
(PARI) A343613_upto(N=100)={localprec(N+5); digits((PrimeZeta32(3)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32.

Crossrefs

Cf. A003627 (primes 3k-1), A085541 (PrimeZeta(3)), A021031 (1/27).

Cf. A175645 (same for p==1 (mod 3)), A086033 (for primes 4k+1), A085992 (for primes 4k+3), A343612 - A343619 (P_{3,2}(2..9): same for 1/p^2, ..., 1/p^9).

Sequence in context: A076942 A205127 A360407 * A333454 A021971 A260822

Adjacent sequences: A343610 A343611 A343612 * A343614 A343615 A343616

Keyword

nonn,cons

Author

M. F. Hasler, Apr 22 2021

Status

approved