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A382636 Decimal expansion of the multiple prime zeta value p[2, 1].

%I A382636 #21 Apr 27 2025 00:46:34
%S A382636 1,5,2,6,6,1,4,1,1,2,5,4,2
%N A382636 Decimal expansion of the multiple prime zeta value p[2, 1].
%C A382636 Prime multiple zeta constants p[m,...,n] are equivalents of multiple zeta constants when successive natural numbers are replaced by successive primes.
%C A382636 For complete list of multiple prime zeta values up to weight 6 see A382234.
%F A382636 Equals p[2, 1] = Sum_{p,q prime p>q} 1/(p^2*q).
%F A382636 Equals p[2, 1] = (p[2, 3] + p[4, 1] + p[2, 1, 2] + 2 p[2, 2, 1])/A085548.
%F A382636 Equals p[2, 1] = sqrt(p[4, 2] + 2 p[2, 2, 2] + 2 p[2, 3, 1] + 2 p[4, 1, 1] + 2 p[2, 1, 2, 1] + 4 p[2, 2, 1, 1]).
%F A382636 A085548*p[2, 1] - p[2, 1, 2] = 0.0531558219243989116479829... [25 digits accuracy].
%F A382636 For partial sums and in infinity occurs identities:
%F A382636 (1) lim_{x->oo} (p[1](x)*A085548 - p[1, 2](x)) = p[2, 1] + A085541 = const.
%F A382636 (2) lim_{x->oo} p[1](x)^3 - 2*p[1](x)*A085548 - p[1, 2](x) - 6*p[1, 1, 1](x) = p[2, 1] - A085541 = const.
%F A382636 (3) lim_{x->oo} (p[1](x)^3 - 3*p[1, 2](x) - 6*p[1, 1, 1](x) = 3*p[2, 1] + A085541 = const.
%F A382636 (4) lim_{x->oo} (p[1](x)^3 - p[1](x)*A085548 - p[1](x)*p[1, 1](x) - p[1, 2](x) - 3*p[1, 1, 1](x)) = p[2, 1] = const.
%F A382636 (5) lim_{x->oo} (p[1](x)*p[1, 1](x) - p[1, 2](x) - 3*p[1, 1, 1](x)) = p[2, 1] = const.
%F A382636 on the left side of each eq. (1)-(5) are divergent series: p[1], p[1, 1], p[1, 2], p[1, 1, 1].
%e A382636 0.1526614112542...
%t A382636 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; p = NextPrime[p], {n, 1, 100000000}], {sum1, n}]
%Y A382636 Cf. A085548, A085541, A085965, A382234, A382235, A382236, A382634, A382635, A382637.
%K A382636 nonn,cons,more
%O A382636 0,2
%A A382636 _Artur Jasinski_, Apr 07 2025