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

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, 14, 56, 29, 10, 31, 2912, 11, 34, 35, 16, 37, 38, 13, 560, 41, 14, 43, 88, 10, 46, 47, 560, 98, 100, 17, 104, 53, 28, 55, 784, 19, 58, 59, 40, 61, 62, 14, 46592, 65

Offset: 1

Author

Vaclav Kotesovec, May 10 2025

Keywords

• nonn
• frac
• mult

Comments

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.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
• Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Crossrefs

Cf. A256688, A256689, A257099, A383705, A383794 (denominators).

Programs

PARI

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-p*X)^(1/3))[n]), ", "))

Formula

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.