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

A330868 Number of proper divisors d of n such that n-d is squarefree.

Original entry on oeis.org

0, 1, 1, 2, 0, 2, 1, 2, 1, 1, 1, 3, 0, 2, 2, 2, 0, 2, 0, 3, 1, 2, 1, 3, 0, 1, 1, 3, 0, 2, 1, 2, 2, 2, 2, 4, 0, 2, 2, 4, 0, 4, 1, 4, 2, 1, 1, 3, 1, 0, 1, 3, 0, 2, 0, 2, 1, 2, 1, 5, 0, 2, 2, 1, 0, 3, 1, 4, 2, 3, 1, 4, 0, 2, 2, 3, 2, 3, 1, 3, 1, 1, 1, 6, 0, 2, 2, 4, 0, 3
Offset: 1

Views

Author

Wesley Ivan Hurt, Apr 28 2020

Keywords

Examples

			a(11) = 1; The only proper divisor of 11 is 1 and 11-1 = 10 is squarefree.
a(12) = 3; There are five proper divisors of 12: 1, 2, 3, 4, 6. Of these, we see that 12-1 = 11, 12-2 = 10 and 12-6 = 6 are squarefree, but 12-3 = 9 and 12-4 = 8 are not.
a(13) = 0; The only proper divisor of 13 is 1, but 13-1 = 12 (which is not squarefree).
a(14) = 2; The proper divisors of 14 are 1, 2, and 7. Of these, only 14-1 = 13 and 14-7 = 7 are squarefree.

Crossrefs

Cf. A001222 (Omega), A007947 (rad), A008683 (Möbius).
Cf. A293227.

Programs

  • Mathematica
    Table[Sum[MoebiusMu[n - i]^2*(1 - Ceiling[n/i] + Floor[n/i]), {i, n - 1}], {n, 100}]
  • PARI
    a(n) = sumdiv(n, d, (dMichel Marcus, Apr 29 2020

Formula

a(n) = Sum_{d|n, dA008683).
Let m = p^k, where p is a prime and k is a positive integer. Then a(p^k) = Sum_{i=0..k-1} mu(p^k - p^i)^2. In terms of m, a(m) = Sum_{j=0..Omega(m)-1} mu(m - rad(m)^j)^2, where mu = A008683, Omega = A001222 and rad = A001222.
Showing 1-3 of 3 results.

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