A256688 -id:A256688 - cp's OEIS frontend

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

Vaclav Kotesovec, May 06 2025

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

Cf. A256689 (denominator).

Cf. A046643, A256688, A383657.

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 *)

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

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.

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.

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

Original entry on oeis.org

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

Vaclav Kotesovec, May 10 2025

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.

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

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

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

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.

A383794 Denominators of Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s-1)^(1/3).

Original entry on oeis.org

1, 3, 1, 9, 3, 3, 3, 81, 1, 9, 3, 9, 3, 9, 3, 243, 3, 3, 3, 27, 3, 9, 3, 81, 9, 9, 3, 27, 3, 9, 3, 729, 3, 9, 9, 9, 3, 9, 3, 243, 3, 9, 3, 27, 3, 9, 3, 243, 9, 27, 3, 27, 3, 9, 9, 243, 3, 9, 3, 27, 3, 9, 3, 6561, 9, 9, 3, 27, 3, 27, 3, 81, 3, 9, 9, 27, 9, 9, 3, 729

Offset: 1

Vaclav Kotesovec, May 10 2025

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

Cf. A256688, A256689, A257099, A383705, A383793 (numerators).

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

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

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

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

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A383794 Denominators of Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s-1)^(1/3).

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