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.

Showing 1-3 of 3 results.

A133699 A051731 * A133698.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 0, 0, 0, 2, 1, 1, 2, 0, 0, 2, 1, 0, 0, 0, 0, 0, 2, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0, 0, 3, 1, 1, 0, 0, 2, 0, 0, 0, 0, 2
Offset: 1

Views

Author

Gary W. Adamson, Sep 21 2007

Keywords

Comments

Right border = A001227, the number of odd divisors of n.
Row sums = A133700: (1, 2, 3, 3, 3, 6, 3, 4, 6, 6, ...)

Examples

			First few rows of the triangle:
  1;
  1, 1;
  1, 0, 2;
  1, 1, 0, 1;
  1, 0, 0, 0, 2;
  1, 1, 2, 0, 0, 2;
  1, 0, 0, 0, 0, 0, 2;
  1, 1, 0, 1, 0, 0, 0, 1;
  ...

Crossrefs

Cf. A051731, A133698, A133700.

Formula

Inverse Mobius transform of A133698, as infinite lower triangular matrices.

A327096 Expansion of Sum_{k>=1} sigma(k) * x^k / (1 - x^(2*k)), where sigma = A000203.

Original entry on oeis.org

1, 3, 5, 7, 7, 15, 9, 15, 18, 21, 13, 35, 15, 27, 35, 31, 19, 54, 21, 49, 45, 39, 25, 75, 38, 45, 58, 63, 31, 105, 33, 63, 65, 57, 63, 126, 39, 63, 75, 105, 43, 135, 45, 91, 126, 75, 49, 155, 66, 114, 95, 105, 55, 174, 91, 135, 105, 93, 61, 245, 63, 99
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 13 2019

Keywords

Comments

Inverse Moebius transform of A002131.
Dirichlet convolution of A000027 with A001227.

Links

Crossrefs

Cf. A000203, A001227, A002131, A007429, A026741, A060640, A109386, A133700, A288417, A300707.

Programs

  • Mathematica
    nmax = 62; CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/(1 - x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := Sum[Total[Select[Divisors[d], OddQ[d/#] &]], {d, Divisors[n]}]; Table[a[n], {n, 1, 62}]
  • PARI
    a(n)={sumdiv(n, d, if(n/d%2, sigma(d)))} \\ Andrew Howroyd, Sep 13 2019

Formula

G.f.: Sum_{k>=1} A002131(k) * x^k / (1 - x^k).
G.f.: Sum_{k>=1} A001227(k) * x^k / (1 - x^k)^2.
a(n) = Sum_{d|n} A002131(d).
a(n) = Sum_{d|n} d * A001227(n/d).
a(n) = (A007429(n) + A288417(n)) / 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/96 = 1.01467803... (A300707). - Amiram Eldar, Oct 23 2022

A326417 Dirichlet g.f.: zeta(s)^4 * (1 - 2^(-s)).

Original entry on oeis.org

1, 3, 4, 6, 4, 12, 4, 10, 10, 12, 4, 24, 4, 12, 16, 15, 4, 30, 4, 24, 16, 12, 4, 40, 10, 12, 20, 24, 4, 48, 4, 21, 16, 12, 16, 60, 4, 12, 16, 40, 4, 48, 4, 24, 40, 12, 4, 60, 10, 30, 16, 24, 4, 60, 16, 40, 16, 12, 4, 96, 4, 12, 40, 28, 16, 48, 4, 24, 16, 48, 4, 100, 4, 12, 40
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 18 2019

Keywords

Comments

Inverse Moebius transform applied twice to A001227.

Links

Crossrefs

Cf. A000005, A001227, A007425, A007426, A133700, A280486, A318845.

Programs

  • Mathematica
    Table[Sum[DivisorSigma[0, n/d] Total[Mod[Divisors[d], 2]], {d, Divisors[n]}], {n, 1, 75}]
nmax = 75; A007425 = Table[DivisorSum[n, DivisorSigma[0, #] &], {n, 1, nmax}]; Table[DivisorSum[n, A007425[[#]] &, OddQ[n/#] &], {n, 1, nmax}]
f[2, e_] := (e + 1)*(e + 2)/2; f[p_, e_] := (e + 1)*(e + 2)*(e + 3)/6; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

Formula

G.f.: Sum_{k>=1} tau_3(k) * x^k / (1 - x^(2*k)), where tau_3 = A007425.
a(n) = tau_4(n) if n odd, tau_4(n) - tau_4(n/2) if n even, where tau_4 = A007426.
a(n) = Sum_{d|n, n/d odd} tau_3(d).
a(n) = Sum_{d|n} A000005(n/d) * A001227(d).
Product_{n>=1} 1 / (1 - x^n)^a(n) = g.f. for A280486.
Multiplicative with a(2^e) = (e+1)*(e+2)/2, and a(p^e) = (e+1)*(e+2)*(e+3)/6 for odd primes p. - Amiram Eldar, Dec 02 2020
Showing 1-3 of 3 results.

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

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