A345320 Sum of the divisors of n whose square does not divide n.

0, 2, 3, 4, 5, 11, 7, 12, 9, 17, 11, 25, 13, 23, 23, 24, 17, 35, 19, 39, 31, 35, 23, 57, 25, 41, 36, 53, 29, 71, 31, 56, 47, 53, 47, 79, 37, 59, 55, 87, 41, 95, 43, 81, 74, 71, 47, 117, 49, 87, 71, 95, 53, 116, 71, 117, 79, 89, 59, 165, 61, 95, 100, 112, 83, 143, 67, 123, 95, 143, 71, 183, 73, 113, 118, 137, 95, 167, 79, 179, 108, 125, 83

Offset

1,2

Links

Table of n, a(n) for n=1..83.

Formula

a(n) = Sum_{k=1..n} k * (ceiling(n/k^2) - floor(n/k^2)) * (1 - ceiling(n/k) + floor(n/k)).

a(n) = A000203(n) - A069290(n). - Rémy Sigrist, Jun 14 2021

Example

a(16) = 24; The divisors of 16 whose square does not divide 16 are 8 and 16. The sum of the divisors is then 8 + 16 = 24.

Mathematica

Table[Sum[k (Ceiling[n/k^2] - Floor[n/k^2]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 80}]

Prog

(PARI) a(n) = sumdiv(n, d, if (n % d^2, d)); \\ Michel Marcus, Jun 13 2021
(Python 3.8+)
from math import prod
from sympy import factorint
def A345320(n):
    f = factorint(n).items()
    return (prod(p**(q+1)-1 for p, q in f) - prod(p**(q//2+1)-1 for p, q in f))//prod(p-1 for p, q in f) # Chai Wah Wu, Jun 14 2021

Crossrefs

Cf. A000203, A056595, A069290.

Sequence in context: A329906 A325651 A345451 * A160000 A245448 A115306

Adjacent sequences: A345317 A345318 A345319 * A345321 A345322 A345323

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 13 2021

Status

approved