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Decimal expansion of C(3), where C(x) = -Sum_{k>=1} (-1)^k/prime(k)^x.
9

%I #46 Oct 30 2022 23:04:25

%S 9,3,4,6,3,6,3,1,3,9,9,6,4,9,8,8,9,1,1,2,4

%N Decimal expansion of C(3), 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(3), after ...91124, seem to converge to a(20)=9, a(21)=0.

%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 instruction.

%e 0.0934636313996498891124...

%t next = 0; ndigits = 11; epsilon = 10^-(2 ndigits); k = 1;

%t While[test = 1/Prime[k + 1]^3 - 1/Prime[k]^3; -test > epsilon,

%t next = next + test; k += 2];

%t First[RealDigits[-next, 10, ndigits]] (* _Robert Price_, Sep 07 2019 *)

%o (PARI) See Sykora link.

%Y Cf. A078437 (x=1), A242301 (x=2), A242303 (x=4), A242304 (x=5).

%Y Cf. A085541.

%K nonn,cons,hard,more

%O -1,1

%A _Stanislav Sykora_, May 14 2014