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

1,2

Links

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

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A000203, A333843.

Sequence in context: A350337 A133477 A303695 * A350339 A039653 A335372

Adjacent sequences: A345318 A345319 A345320 * A345322 A345323 A345324

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 13 2021

Status

approved