cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 21-26 of 26 results.

A257100 From fourth root of the inverse of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose fourth power is 1/zeta; sequence gives numerator of b(n).

Original entry on oeis.org

1, -1, -1, -3, -1, 1, -1, -7, -3, 1, -1, 3, -1, 1, 1, -77, -1, 3, -1, 3, 1, 1, -1, 7, -3, 1, -7, 3, -1, -1, -1, -231, 1, 1, 1, 9, -1, 1, 1, 7, -1, -1, -1, 3, 3, 1, -1, 77, -3, 3, 1, 3, -1, 7, 1, 7, 1, 1, -1, -3, -1, 1, 3, -1463, 1, -1, -1, 3, 1, -1, -1, 21, -1, 1, 3, 3, 1, -1, -1, 77, -77, 1, -1, -3, 1, 1, 1, 7, -1, -3, 1, 3, 1, 1, 1, 231, -1, 3, 3, 9
Offset: 1

Views

Author

Wolfgang Hintze, Apr 16 2015

Keywords

Comments

Dirichlet g.f. of b(n) = a(n)/A256691(n) is (zeta(x))^(-1/4).
Denominator is the same as for Dirichlet g.f. (zeta(x))^(+1/4).
Formula holds for general Dirichlet g.f. zeta(x)^(-1/k) with k = 1, 2, ...

Links

Crossrefs

Cf. family zeta^(-1/k): A257098/A046644 (k=2), A257099/A256689 (k=3), A257100/A256691 (k=4), A257101/A256693 (k=5).
Cf. family zeta^(+1/k): A046643/A046644 (k=2), A256688/A256689 (k=3), A256690/A256691 (k=4), A256692/A256693 (k=5).

Programs

  • Mathematica
    k = 4;
c[1, n_] = b[n];
c[k_, n_] := DivisorSum[n, c[1, #1]*c[k - 1, n/#1] & ]
nn = 100; eqs = Table[c[k, n]==MoebiusMu[n], {n, 1, nn}];
sol = Solve[Join[{b[1] == 1}, eqs], Table[b[i], {i, 1, nn}], Reals];
t = Table[b[n], {n, 1, nn}] /. sol[[1]];
num = Numerator[t] (* A257100 *)
den = Denominator[t] (* A256691 *)
  • PARI
    for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^(-1/4))[n]), ", ")) \\ Vaclav Kotesovec, May 04 2025

Formula

with k = 4;
zeta(x)^(-1/k) = Sum_{n>=1} b(n)/n^x;
c(1,n)=b(n); c(k,n) = Sum_{d|n} c(1,d)*c(k-1,n/d), k>1;
Then solve c(k,n) = mu(n) for b(m);
a(n) = numerator(b(n)).
Sum_{j=1..n} A257100(j)/A256691(j) ~ n / (Gamma(-1/4) * log(n)^(5/4)) * (1 + 5*(gamma/4 + 1)/(4*log(n))), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the gamma function. - Vaclav Kotesovec, May 05 2025

A257101 From fifth root of the inverse of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose fifth power is 1/zeta; sequence gives numerator of b(n).

Original entry on oeis.org

1, -1, -1, -2, -1, 1, -1, -6, -2, 1, -1, 2, -1, 1, 1, -21, -1, 2, -1, 2, 1, 1, -1, 6, -2, 1, -6, 2, -1, -1, -1, -399, 1, 1, 1, 4, -1, 1, 1, 6, -1, -1, -1, 2, 2, 1, -1, 21, -2, 2, 1, 2, -1, 6, 1, 6, 1, 1, -1, -2, -1, 1, 2, -1596, 1, -1, -1, 2, 1, -1, -1, 12, -1, 1, 2, 2, 1, -1, -1, 21, -21, 1, -1, -2, 1, 1, 1, 6, -1, -2, 1, 2, 1, 1, 1, 399, -1, 2, 2, 4
Offset: 1

Views

Author

Wolfgang Hintze, Apr 16 2015

Keywords

Comments

Dirichlet g.f. of b(n) = A257101(n)/A256693(n) is (zeta(x))^(-1/5).
Denominator is the same as for Dirichlet g.f. (zeta(x))^(+1/5).
Formula holds for general Dirichlet g.f. zeta(x)^(-1/k) with k = 1, 2, ...

Links

Crossrefs

Cf. family zeta^(-1/k): A257098/A046644 (k=2), A257099/A256689 (k=3), A257100/A256691 (k=4), A257101/A256693 (k=5).
Cf. family zeta^(+1/k): A046643/A046644 (k=2), A256688/A256689 (k=3), A256690/A256691 (k=4), A256692/A256693 (k=5).

Programs

  • Mathematica
    k = 5;
c[1, n_] = b[n];
c[k_, n_] := DivisorSum[n, c[1, #1]*c[k - 1, n/#1] & ]
nn = 100; eqs = Table[c[k, n]==MoebiusMu[n], {n, 1, nn}];
sol = Solve[Join[{b[1] == 1}, eqs], Table[b[i], {i, 1, nn}], Reals];
t = Table[b[n], {n, 1, nn}] /. sol[[1]];
num = Numerator[t] (* A257101 *)
den = Denominator[t] (* A256693 *)
  • PARI
    for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^(-1/5))[n]), ", ")) \\ Vaclav Kotesovec, May 04 2025

Formula

with k = 5;
zeta(x)^(-1/k) = Sum_{n>=1} b(n)/n^x;
c(1,n)=b(n); c(k,n) = Sum_{d|n} c(1,d)*c(k-1,n/d), k>1;
Then solve c(k,n) = mu(n) for b(m);
a(n) = numerator(b(n)).
Sum_{j=1..n} A257101(j)/A256693(j) ~ n / (Gamma(-1/5) * log(n)^(6/5)) * (1 + 6*(gamma/5 + 1)/(5*log(n))), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the gamma function. - Vaclav Kotesovec, May 05 2025

A317848 Multiplicative with a(p^e) = binomial(2*e, e).

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 20, 6, 4, 2, 12, 2, 4, 4, 70, 2, 12, 2, 12, 4, 4, 2, 40, 6, 4, 20, 12, 2, 8, 2, 252, 4, 4, 4, 36, 2, 4, 4, 40, 2, 8, 2, 12, 12, 4, 2, 140, 6, 12, 4, 12, 2, 40, 4, 40, 4, 4, 2, 24, 2, 4, 12, 924, 4, 8, 2, 12, 4, 8, 2, 120, 2, 4, 12, 12, 4, 8, 2, 140
Offset: 1

Views

Author

Andrew Howroyd, Aug 08 2018

Keywords

Comments

The Dirichlet convolution square of this sequence is A165825.

Links

Crossrefs

Cf. A000265, A000984, A006519, A037445, A046643, A046644, A165825, A299149, A299150.

Programs

  • Mathematica
    f[p_, e_] := Binomial[2*e, e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 30 2023 *)
  • PARI
    a(n)={my(v=factor(n)[,2]); prod(i=1, #v, binomial(2*v[i], v[i]))}
  • PARI
    \\ DirSqrt(v) finds u such that v = v[1]*dirmul(u, u).
DirSqrt(v)={my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&d
  • PARI
    A317848(n) = factorback(apply(e -> binomial(e+e,e),factor(n)[,2])); \\ Antti Karttunen, Sep 17 2018

Formula

A037445(n) = A006519(a(n)).
A046643(n) = numerator(a(n)/A165825(n)) = A000265(a(n)).
A046644(n) = denominator(a(n)/A165825(n)) = A165825(n)/A037445(n).
A299149(n) = numerator(n*a(n)/A165825(n)) = A000265(n*a(n)).
A299150(n) = denominator(n*a(n)/A165825(n)) = A165825(n)/(A037445(n) * A006519(n)).

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

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, May 04 2025

Keywords

Comments

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.

Links

Crossrefs

Cf. A383658.
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]],Numerator[Times@@(coeff[[flist[[All,2]]+1]])]]];Array[dptTerm,73] (* Shenghui Yang, May 04 2025 *)
  • PARI
    for(n=1, 100, print1(numerator(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.

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

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, May 06 2025

Keywords

Links

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.

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

Views

Author

Vaclav Kotesovec, May 04 2025

Keywords

Comments

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

Links

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.
Previous Showing 21-26 of 26 results.

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