login
A345321
Sum of the divisors of n whose cube does not divide n.
1
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
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
Sequence in context: A350337 A133477 A303695 * A350339 A039653 A335372
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 13 2021
STATUS
approved