A256693 -id:A256693 - cp's OEIS frontend

A383657 Numerator of Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s)^(3/2).

1, 3, 3, 15, 3, 9, 3, 35, 15, 9, 3, 45, 3, 9, 9, 315, 3, 45, 3, 45, 9, 9, 3, 105, 15, 9, 35, 45, 3, 27, 3, 693, 9, 9, 9, 225, 3, 9, 9, 105, 3, 27, 3, 45, 45, 9, 3, 945, 15, 45, 9, 45, 3, 105, 9, 105, 9, 9, 3, 135, 3, 9, 45, 3003, 9, 27, 3, 45, 9, 27, 3, 525, 3

Offset: 1

Vaclav Kotesovec, May 04 2025

In general, for m > 0, if Dirichlet g.f. is zeta(s)^m, then Sum_{j=1..n} a(j) ~ n*log(n)^(m-1)/Gamma(m) * (1 + (m-1)*(m*gamma - 1)/log(n)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the gamma function.

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

Cf. A383658.
Cf. A046643, A046644, A256688, A256689, A256690, A256691, A256692, A256693.
Cf. A000005, A007425, A007426, A061200, A034695.

coeff=CoefficientList[Series[1/(1-x)^(3/2),{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,73] (* Shenghui Yang, May 04 2025 *)

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

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.

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

Original entry on oeis.org

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

Vaclav Kotesovec, May 04 2025

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

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

Cf. A383657.
Cf. A046643, A046644, A256688, A256689, A256690, A256691, A256692, A256693.
Cf. A000005, A007425, A007426, A061200, A034695.

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 *)

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

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.

cp's OEIS Frontend

A383657 Numerator of Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s)^(3/2).

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