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.

A293228 a(n) is the sum of proper divisors of n that are squarefree.

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 3, 4, 8, 1, 12, 1, 10, 9, 3, 1, 12, 1, 18, 11, 14, 1, 12, 6, 16, 4, 24, 1, 42, 1, 3, 15, 20, 13, 12, 1, 22, 17, 18, 1, 54, 1, 36, 24, 26, 1, 12, 8, 18, 21, 42, 1, 12, 17, 24, 23, 32, 1, 72, 1, 34, 32, 3, 19, 78, 1, 54, 27, 74, 1, 12, 1, 40, 24, 60, 19, 90, 1, 18, 4, 44, 1, 96, 23, 46, 33, 36, 1, 72, 21, 72, 35, 50, 25, 12, 1, 24
Offset: 1

Views

Author

Antti Karttunen, Oct 08 2017

Keywords

Links

Crossrefs

Cf. A005117, A008966, A048250, A293227, A293235.

Programs

  • Maple
    with(numtheory): seq(coeff(series(add(mobius(k)^2*k*x^(2*k)/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 120); # Muniru A Asiru, Oct 28 2018
  • Mathematica
    a[n_] := Times @@ (1 + (f = FactorInteger[n])[[;; , 1]]) - If[AllTrue[f[[;;, 2]], # == 1 &], n, 0]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Oct 09 2022 *)
Table[Total[Select[Most[Divisors[n]],SquareFreeQ]],{n,100}] (* Harvey P. Dale, Apr 20 2025 *)
  • PARI
    A293228(n) = sumdiv(n, d, (d

Formula

a(n) = Sum_{d|n, dA008966(d)*d.
a(n) = A048250(n) - (A008966(n)*n).
G.f.: Sum_{k>=1} mu(k)^2*k*x^(2*k)/(1 - x^k). - Ilya Gutkovskiy, Oct 28 2018
From Amiram Eldar, Oct 09 2022: (Start)
a(n) = 1 iff n is a prime.
a(n) = 3 iff n is a power of 2 greater than 2 (A020707).
Sum_{k=1..n} a(k) ~ (1/2 - 3/Pi^2) * n^2. (End)

A293234 a(n) is the number of proper divisors of n that are square.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 2, 2, 3, 1, 1, 1, 2, 1
Offset: 1

Views

Author

Antti Karttunen, Oct 08 2017

Keywords

Comments

First occurrence of k: 2, 8, 32, 72, 144, 288, 576, 1152, 2304, 4608, 3600, 7200, 36864, 20736, 14400, 28800, 32400, 64800, 57600, 115200, 663552, 18874368, 129600, 259200, 3359232, 810000, 921600, 1843200, 518400, 1036800, 705600, 1411200, etc. - Robert G. Wilson v, Oct 08 2017

Links

Crossrefs

Cf. A010052, A013661, A046951, A293227, A293235.

Programs

  • Mathematica
    f[n_] := Length@ Select[ Sqrt@ Most@ Divisors@ n, IntegerQ]; Array[f, 105] (* Robert G. Wilson v, Oct 08 2017 *)
  • PARI
    A293234(n) = sumdiv(n,d,(d

Formula

a(n) = Sum_{d|n, dA010052(d).
a(n) = A046951(n) - A010052(n).
G.f.: Sum_{k>=1} x^(2*k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Apr 13 2021
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi^2/6 (A013661). - Amiram Eldar, Dec 01 2023

A343408 Sum of proper divisors of n that are triangular numbers.

Original entry on oeis.org

0, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 10, 1, 1, 4, 1, 1, 10, 1, 11, 4, 1, 1, 10, 1, 1, 4, 1, 1, 35, 1, 1, 4, 1, 1, 10, 1, 1, 4, 11, 1, 31, 1, 1, 19, 1, 1, 10, 1, 11, 4, 1, 1, 10, 1, 29, 4, 1, 1, 35, 1, 1, 25, 1, 1, 10, 1, 1, 4, 11, 1, 46, 1, 1, 19, 1, 1, 10, 1, 11, 4, 1, 1, 59, 1, 1, 4, 1, 1, 80, 1, 1, 4, 1, 1, 10
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 14 2021

Keywords

Crossrefs

Cf. A000217, A001065, A010054, A185027, A293235, A343407.

Programs

  • Maple
    a:= n-> add(`if`(issqr(8*d+1), d, 0), d=numtheory[divisors](n) minus {n}):
seq(a(n), n = 1..96);  # Alois P. Heinz, Apr 14 2021
  • Mathematica
    nmax = 96; CoefficientList[Series[Sum[(k (k + 1)/2) x^(k (k + 1))/(1 - x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[If[d < n && IntegerQ[Sqrt[8 d + 1]], d, 0], {d, Divisors[n]}], {n, 96}]
  • PARI
    a(n) = sumdiv(n, d, if ((dMichel Marcus, Apr 14 2021

Formula

G.f.: Sum_{k>=1} (k*(k+1)/2) * x^(k*(k+1)) / (1 - x^(k*(k+1)/2)).
a(n) = Sum_{d|n, d < n} A010054(d) * d.
Showing 1-3 of 3 results.

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