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

1,4

Links

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

Formula

a(n) = Sum_{d|n, d<n} A008966(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)

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 *)

Prog

(PARI) A293228(n) = sumdiv(n, d, (d<n)*issquarefree(d)*d);

Crossrefs

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

Sequence in context: A115364 A016476 A293436 * A169814 A068436 A019570

Adjacent sequences: A293225 A293226 A293227 * A293229 A293230 A293231

Keyword

nonn

Author

Antti Karttunen, Oct 08 2017

Status

approved