A345321 Sum of the divisors of n whose cube does not divide n.

0, 2, 3, 6, 5, 11, 7, 12, 12, 17, 11, 27, 13, 23, 23, 28, 17, 38, 19, 41, 31, 35, 23, 57, 30, 41, 36, 55, 29, 71, 31, 60, 47, 53, 47, 90, 37, 59, 55, 87, 41, 95, 43, 83, 77, 71, 47, 121, 56, 92, 71, 97, 53, 116, 71, 117, 79, 89, 59, 167, 61, 95, 103, 120, 83, 143, 67, 125, 95

Offset: 1

Author

Wesley Ivan Hurt, Jun 13 2021

Keywords

• nonn

Examples

			a(16) = 28; The divisors of 16 whose cube does not divide 16 are: 4, 8 and 16. The sum of these divisors is then 4 + 8 + 16 = 28.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000203, A333843.

Programs

Mathematica

Table[Sum[k (Ceiling[n/k^3] - Floor[n/k^3]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 80}]
Table[Total[Select[Divisors[n],Mod[n,#^3]!=0&]],{n,100}] (* Harvey P. Dale, May 01 2022 *)

PARI

a(n) = sumdiv(n, d, if (n % d^3, d)); \\ Michel Marcus, Jun 13 2021
(Python 3.8+)
from math import prod
from sympy import factorint
def A345321(n):
    f = factorint(n).items()
    return prod((p**(q+1)-1)//(p-1) for p, q in f) - prod((p**(q//3+1)-1)//(p-1) for p, q in f) # Chai Wah Wu, Jun 14 2021

Formula

a(n) = Sum_{k=1..n} k * (ceiling(n/k^3) - floor(n/k^3)) * (1 - ceiling(n/k) + floor(n/k)).

a(n) = A000203(n) - A333843(n). - Rémy Sigrist, Jun 14 2021