A242301 - OEIS

A242301

Decimal expansion of C(2), where C(x) = -Sum{k>=1} (-1)^k/prime(k)^x.

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

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OFFSET

0,2

COMMENTS

The alternating series of reciprocal powers of prime numbers converges for any x > 0 (absolutely so if x > 1) but is hard to compute.

The next digits of C(2), after ...6014, seem to converge to a(16)=1, a(17)=5.

LINKS

Table of n, a(n) for n=0..22.

Martin Fuller, C program

Stanislav Sykora, PARI/GP scripts for primes-related functions, see function AltSum1DivPrimePwr(x,eps), with instructions.

Eric Weisstein's World of Mathematics, Prime Sums

Eric Weisstein's World of Mathematics, Prime Zeta Function

Wikipedia, Prime Zeta Function

EXAMPLE

0.16281624666360141576833...

MATHEMATICA

k = 1; p = 2; s = 0; While[p < 1000000000, s = N[s + (-1)^k/p^2, 40]; k = Mod[++k, 2]; p = NextPrime@ p]; s (* takes ~30 minutes on an average laptop to 18 decimal digits *)(* Robert G. Wilson v, Dec 30 2017 *)

PROG

(PARI) See Sykora link.

CROSSREFS

Cf. A078437 (x=1), A242302 (x=3), A242303 (x=4), A242304 (x=5).

Cf. A085548.

Sequence in context: A177889 A388635 A086744 * A256129 A385495 A019692

Adjacent sequences: A242298 A242299 A242300 * A242302 A242303 A242304

KEYWORD

nonn,cons,hard,more

AUTHOR

Stanislav Sykora, May 14 2014

EXTENSIONS

a(16)-a(22) from Martin Fuller, Apr 22 2026

STATUS

approved