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.

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A379362 Denominators of the partial alternating sums of the reciprocals of the number of abelian groups function (A000688).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 6, 3, 3, 3, 6, 6, 6, 6, 30, 30, 15, 15, 30, 30, 30, 30, 30, 15, 15, 15, 30, 30, 30, 30, 210, 210, 210, 210, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 140, 140, 420, 420, 420, 420, 420, 420, 420
Offset: 1

Views

Author

Amiram Eldar, Dec 21 2024

Keywords

Links

Crossrefs

Cf. A000688, A063966, A370897, A379360, A379361 (numerators).

Programs

  • Mathematica
    Denominator[Accumulate[Table[(-1)^(n+1)/FiniteAbelianGroupCount[n], {n, 1, 100}]]]
  • PARI
    f(n) = vecprod(apply(numbpart, factor(n)[, 2]));
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / f(k); print1(denominator(s), ", "))};

Formula

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/A000688(k)).

A328705 Dirichlet g.f.: Product_{k>=1} zeta(k*s)^2.

Original entry on oeis.org

1, 2, 2, 5, 2, 4, 2, 10, 5, 4, 2, 10, 2, 4, 4, 20, 2, 10, 2, 10, 4, 4, 2, 20, 5, 4, 10, 10, 2, 8, 2, 36, 4, 4, 4, 25, 2, 4, 4, 20, 2, 8, 2, 10, 10, 4, 2, 40, 5, 10, 4, 10, 2, 20, 4, 20, 4, 4, 2, 20, 2, 4, 10, 65, 4, 8, 2, 10, 4, 8, 2, 50, 2, 4, 10, 10, 4, 8, 2, 40
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 26 2019

Keywords

Comments

Dirichlet convolution of A000688 with itself.

Links

Crossrefs

Cf. A000688, A000712, A001620, A063966, A129667, A188581, A188585, A301830.

Programs

  • Mathematica
    Table[DivisorSum[n, FiniteAbelianGroupCount[n/#] FiniteAbelianGroupCount[#] &], {n, 1, 80}]

Formula

a(n) = Sum_{d|n} A000688(n/d) * A000688(d).
Sum_{k=1..n} a(k) ~ c^2 * n * (log(n) + 2*gamma - 1 - 2*s), where c = A021002 = Product_{k>=2} zeta(k) = 2.2948565916733137941835158313443112887131637994..., s = Sum_{k>=2} k*zeta'(k)/zeta(k) = -2.1955691982567064617939038695473479681910375... and gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 26 2019
Multiplicative with a(p^e) = A000712(e). - Amiram Eldar, Nov 30 2020

A358780 Dirichlet g.f.: zeta(s) * zeta(2*s) * zeta(3*s) * zeta(4*s).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 9, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1, 3
Offset: 1

Views

Author

Vaclav Kotesovec, Mar 14 2023

Keywords

Comments

a(n) = A000688(n) for n < 32.

Links

Crossrefs

Cf. A046951, A322885, A000688.
Cf. A001400, A063966.

Programs

  • PARI
    for(n=1, 200, print1(direuler(p=2, n, 1/(1 - X)/(1 - X^2)/(1 - X^3)/(1 - X^4))[n], ", "))

Formula

Multiplicative with a(p^e) = A001400(e).
Sum_{k=1..n} a(k) ~ Pi^6*zeta(3)*n/540 + Pi^2*zeta(1/2)*zeta(3/2)*sqrt(n)/6 + zeta(1/3)*zeta(2/3)*zeta(4/3)*n^(1/3) + zeta(1/4)*zeta(1/2)*zeta(3/4)*n^(1/4).
Previous Showing 11-13 of 13 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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