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A383793 Numerators of Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s-1)^(1/3).

%I A383793 #13 May 11 2025 07:38:19
%S A383793 1,2,1,8,5,2,7,112,2,10,11,8,13,14,5,560,17,4,19,40,7,22,23,112,50,26,
%T A383793 14,56,29,10,31,2912,11,34,35,16,37,38,13,560,41,14,43,88,10,46,47,
%U A383793 560,98,100,17,104,53,28,55,784,19,58,59,40,61,62,14,46592,65
%N A383793 Numerators of Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s-1)^(1/3).
%C A383793 General formula: if k >= 0, m > 0, and the Dirichlet generating function is zeta(s-k)^m * f(s), where f(s) has all possible poles at points less than k+1, then Sum_{j=1..n} a(j) ~ n^(k+1) * log(n)^(m-1) * f(k+1) / ((k+1) * Gamma(m)) * (1 + (m-1)*(m*gamma - 1/(k+1) + f'(k+1)/f(k+1)) / log(n)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.
%H A383793 Vaclav Kotesovec, <a href="/A383793/b383793.txt">Table of n, a(n) for n = 1..10000</a>
%H A383793 Vaclav Kotesovec, <a href="/A383793/a383793.jpg">Graph - the asymptotic ratio (10000 terms)</a>
%F A383793 Sum_{k=1..n} A383793(k) / A383794(k) ~ n^2 / (2*Gamma(1/3)*log(n)^(2/3)) * (1 + (1 - 2*gamma/3)/(3*log(n))), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.
%o A383793 (PARI) for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-p*X)^(1/3))[n]), ", "))
%Y A383793 Cf. A256688, A256689, A257099, A383705, A383794 (denominators).
%K A383793 nonn,frac,mult
%O A383793 1,2
%A A383793 _Vaclav Kotesovec_, May 10 2025