cp's OEIS Frontend

A382637 Decimal expansion of the multiple prime zeta value p[3, 1].

%I A382637 #17 May 03 2025 21:56:20
%S A382637 0,3,0,5,3,1,1,6,4,0,5,7,9,4
%N A382637 Decimal expansion of the multiple prime zeta value p[3, 1].
%C A382637 Prime multiple zeta constants p[m,...,n] are equivalents of multiple zeta constants when successive natural numbers are replaced by successive primes.
%C A382637 For complete list of multiple prime zeta values up to weight 6 see A382234.
%F A382637 Equals Sum_{p,q prime p>q} 1/(p^3*q).
%F A382637 p[3, 1] + p[4] = lim_{x->oo}  p[3]*p[1](x) - p[1, 3](x) = 0.1075243038221868... = cons, where p[1] and p[1, 3] are divergent series.
%F A382637 p[3, 1] + p[2, 2] + 2 p[2, 1, 1] = lim_{x->oo} p[2, 1]*p[1](x) - p[1, 2, 1](x) = 0.1686331457227234... = cons, where p[1] and p[2, 1, 1] are divergent series.
%e A382637 0.03053116405794...
%t A382637 p3 = N[PrimeZetaP[3], 50]; p = 2; sum = 0; sum1 = 0; diff = 0; Monitor[
%t A382637  Do[sum = sum + N[1/p^3, 50]; diff = p3 - sum; sum1 = sum1 + diff/p;
%t A382637   p = NextPrime[p], {n, 1, 100000000}], {sum1, n}]
%o A382637 (PARI)
%o A382637 f(e)=my(S=sumeulerrat(1/x^3), u=0., v=0); forprime(p=2, 2^e, u+=v*S; S-=1/p^3; v=1/p); u;f(30) // Bill Allombert
%Y A382637 Cf. A085548, A085541, A085965, A382234, A382235, A382236, A382634, A382635, A382636, A383432.
%K A382637 nonn,cons,more
%O A382637 0,2
%A A382637 _Artur Jasinski_, Apr 27 2025