|
|
A162296
|
|
Sum of divisors of n that have a square factor.
|
|
40
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,4
|
|
COMMENTS
|
|
|
LINKS
|
|
|
FORMULA
|
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
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
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|