A162296 - OEIS

%I #33 Apr 20 2023 14:55:16

%S 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,

%T 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,

%U 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

%N Sum of divisors of n that have a square factor.

%C Note that 1 does not have a square factor. - _Antti Karttunen_, Nov 20 2017

%H Antti Karttunen, <a href="/A162296/b162296.txt">Table of n, a(n) for n = 1..16384</a>

%H <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>.

%F a(n) + A048250(n) = A000203(n). - _Antti Karttunen_, Nov 20 2017

%F From _Amiram Eldar_, Oct 01 2022: (Start)

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

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

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

%e a(8) = 12 = 4 + 8.

%t Array[DivisorSum[#, # &, # (1 - MoebiusMu[#]^2) == # &] &, 86] (* _Michael De Vlieger_, Nov 20 2017 *)

%t a[1]=0; a[n_] := DivisorSigma[1, n] - Times@@(1+FactorInteger[n][[;; , 1]]); Array[a,86] (* _Amiram Eldar_, Dec 20 2018 *)

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

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A162296(n):

%o     f = factorint(n)

%o     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

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

%K easy,nonn

%O 1,4

%A _Joerg Arndt_, Jun 30 2009