cp's OEIS Frontend

A382634 Decimal expansion of the multiple prime zeta value p[2, 3].

%I A382634 #29 Apr 27 2025 00:45:31
%S A382634 0,2,9,1,8,5,1,6,6,5,0,4,0,1,2,5,7,1,0,4,0,6,4,1,3,4,4,0,9,0,2,7,9,1,
%T A382634 9,6,7,4,7
%N A382634 Decimal expansion of the multiple prime zeta value p[2, 3].
%C A382634 Prime multiple zeta constants p[m,...,n] are equivalents of multiple zeta constants when successive natural numbers are replaced by successive primes.
%C A382634 For complete list of multiple prime zeta values up to weight 6 see A382234.
%F A382634 Equals Sum_{p,q prime p>q} 1/(p^2*q^3).
%F A382634 Equals A085548*A085541 - A085965 - A382635.
%F A382634 For partial sums and in infinity occurs identity:
%F A382634 p[2, 3] + p[3, 2] + p[2, 1, 2] + p[2, 2, 1] = p[1] p[2, 2]- p[1, 2, 2]
%F A382634 where p[1] and p[1, 2, 2] are divergent series then
%F A382634 lim_{n->oo} p[1](n)*A382234 - p[1, 2, 2](n) = 0.067101047034256...
%e A382634 0.029185166504012571040641344090279196747...
%t A382634 p2 = N[PrimeZetaP[2], 50]; p = 2; sum = 0; sum1 = 0; diff = 0; Monitor[Do[sum = sum + N[1/p^2, 50]; diff = p2 - sum; sum1 = sum1 + diff/p^3; p = NextPrime[p], {n, 1, 100000000}], {sum1, n}]
%Y A382634 Cf. A085548, A085541, A085965, A382234, A382235, A382236, A382635.
%Y A382634 Cf. A258986.
%K A382634 nonn,cons,more
%O A382634 0,2
%A A382634 _Artur Jasinski_, Apr 01 2025