A162296 - OEIS

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

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OFFSET

1,4

COMMENTS

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

LINKS

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

Index entries for sequences related to sums of divisors.

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)

EXAMPLE

a(8) = 12 = 4 + 8.

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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A126849 A284117 A183099 * A364103 A169773 A236380

Adjacent sequences: A162293 A162294 A162295 * A162297 A162298 A162299

KEYWORD

easy,nonn

AUTHOR

Joerg Arndt, Jun 30 2009

STATUS

approved