A383658 Denominator of Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s)^(3/2).

1, 2, 2, 8, 2, 4, 2, 16, 8, 4, 2, 16, 2, 4, 4, 128, 2, 16, 2, 16, 4, 4, 2, 32, 8, 4, 16, 16, 2, 8, 2, 256, 4, 4, 4, 64, 2, 4, 4, 32, 2, 8, 2, 16, 16, 4, 2, 256, 8, 16, 4, 16, 2, 32, 4, 32, 4, 4, 2, 32, 2, 4, 16, 1024, 4, 8, 2, 16, 4, 8, 2, 128, 2, 4, 16, 16, 4

Offset: 1

Author

Vaclav Kotesovec, May 04 2025

Keywords

• nonn
• frac
• mult

Comments

Is this a duplicate of A046644 (the first 8192 entries are the same)? - R. J. Mathar, May 06 2025

Links

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

Crossrefs

Cf. A383657.

Cf. A046643, A046644, A256688, A256689, A256690, A256691, A256692, A256693.

Cf. A000005, A007425, A007426, A061200, A034695.

Programs

Mathematica

coeff=CoefficientList[Series[1/(1-x)^(3/2),{x,0,20}]//Normal,x]; dptTerm[n_]:=Module[{flist=FactorInteger[n]},If[n==1,coeff[[1]],Denominator[Times@@(coeff[[flist[[All,2]]+1]])]]];Array[dptTerm,77] (* Shenghui Yang, May 04 2025 *)

PARI

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

Formula

Sum_{k=1..n} A383657(k)/A383658(k) ~ 2*n*sqrt(log(n)/Pi) * (1 - (1 - 3*gamma/2)/(2*log(n))), where gamma is the Euler-Mascheroni constant A001620.