A383705 Numerator of Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s)^(2/3).

1, 2, 2, 5, 2, 4, 2, 40, 5, 4, 2, 10, 2, 4, 4, 110, 2, 10, 2, 10, 4, 4, 2, 80, 5, 4, 40, 10, 2, 8, 2, 308, 4, 4, 4, 25, 2, 4, 4, 80, 2, 8, 2, 10, 10, 4, 2, 220, 5, 10, 4, 10, 2, 80, 4, 80, 4, 4, 2, 20, 2, 4, 10, 2618, 4, 8, 2, 10, 4, 8, 2, 200, 2, 4, 10, 10, 4

Offset: 1

Author

Vaclav Kotesovec, May 06 2025

Keywords

• nonn
• frac
• mult

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A256689 (denominator).

Cf. A046643, A256688, A383657.

Programs

Mathematica

coeff=CoefficientList[Series[1/(1-x)^(2/3),{x,0,20}]//Normal,x];dptTerm[n_]:=Module[{flist=FactorInteger[n]},If[n==1,coeff[[1]],Numerator[Times@@(coeff[[flist[[All,2]]+1]])]]];Array[dptTerm,77] (* Shenghui Yang, May 06 2025 *)

PARI

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

Formula

Sum_{k=1..n} a(k)/A256689(k) ~ n / (Gamma(2/3) * log(n)^(1/3)) * (1 + (1 - 2*gamma/3)/(3*log(n))), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the gamma function.