A162296 Sum of divisors of n that have a square factor.

0, 0, 0, 4, 0, 0, 0, 12, 9, 0, 0, 16, 0, 0, 0, 28, 0, 27, 0, 24, 0, 0, 0, 48, 25, 0, 36, 32, 0, 0, 0, 60, 0, 0, 0, 79, 0, 0, 0, 72, 0, 0, 0, 48, 54, 0, 0, 112, 49, 75, 0, 56, 0, 108, 0, 96, 0, 0, 0, 96, 0, 0, 72, 124, 0, 0, 0, 72, 0, 0, 0, 183, 0, 0, 100, 80, 0, 0, 0, 168, 117, 0, 0, 128, 0, 0

Offset: 1

Author

Joerg Arndt, Jun 30 2009

Keywords

• easy
• nonn

Comments

Note that 1 does not have a square factor. - Antti Karttunen, Nov 20 2017

Examples

			a(8) = 12 = 4 + 8.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences related to sums of divisors.

Crossrefs

Cf. A000203, A001248, A005117, A013929, A048250.

Programs

Mathematica

Array[DivisorSum[#, # &, # (1 - MoebiusMu[#]^2) == # &] &, 86] (* Michael De Vlieger, Nov 20 2017 *)
a[1]=0; a[n_] := DivisorSigma[1, n] - Times@@(1+FactorInteger[n][[;; , 1]]); Array[a,86] (* Amiram Eldar, Dec 20 2018 *)

PARI

a(n)=sumdiv(n,d,d*(1-moebius(d)^2)); v=vector(300,n,a(n))

Python

from math import prod
from sympy import factorint
def A162296(n):
    f = factorint(n)
    return prod((p**(e+1)-1)//(p-1) for p, e in f.items())-prod(p+1 for p in f) # Chai Wah Wu, Apr 20 2023

Formula

a(n) + A048250(n) = A000203(n). - Antti Karttunen, Nov 20 2017

From Amiram Eldar, Oct 01 2022: (Start)

a(n) = 0 iff n is squarefree (A005117).

a(n) = n iff n is a square of a prime (A001248).

Sum_{k=1..n} a(k) ~ (Pi^2/12 - 1/2) * n^2. (End)