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

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

Author

Antti Karttunen, Oct 08 2017

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

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)