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Decimal expansion of C(2), where C(x) = -Sum{k>=1} (-1)^k/prime(k)^x.
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%I #46 Oct 30 2022 23:04:20

%S 1,6,2,8,1,6,2,4,6,6,6,3,6,0,1,4

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

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

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

%H Stanislav Sykora, <a href="https://oeis.org/wiki/File:PrimesRelatedFunctions.txt">PARI/GP scripts for primes-related functions</a>, see function AltSum1DivPrimePwr(x,eps), with instructions.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeSums.html">Prime Sums</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeZetaFunction.html">Prime Zeta Function</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime_zeta_function">Prime Zeta Function</a>

%e 0.1628162466636014...

%t 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 *)

%o (PARI) See Sykora link.

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

%Y Cf. A085548.

%K nonn,cons,hard,more

%O 0,2

%A _Stanislav Sykora_, May 14 2014