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

Author

Wesley Ivan Hurt, Jun 13 2021

Keywords

• nonn

Comments

Inverse Möbius transform of n+n^(1/2)*((-1)^tau(n)-1)/2. - Wesley Ivan Hurt, Jul 07 2025

Examples

			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.
		

Links

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

Crossrefs

Cf. A000005 (tau), A000203 (sigma), A056595, A069290.

Programs

Mathematica

Table[Sum[k (Ceiling[n/k^2] - Floor[n/k^2]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 80}]
sdnd[n_]:=Total[Select[Divisors[n],Mod[n,#^2]!=0&]]; Array[sdnd,100] (* Harvey P. Dale, Jul 07 2025 *)

PARI

a(n) = sumdiv(n, d, if (n % d^2, d)); \\ Michel Marcus, Jun 13 2021

Python

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

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

a(n) = Sum_{d|n} (d+d^(1/2)*((-1)^tau(d)-1)/2). - Wesley Ivan Hurt, Jul 07 2025