A320059 Sum of divisors of n^2 that do not divide n.

0, 4, 9, 24, 25, 79, 49, 112, 108, 199, 121, 375, 169, 375, 379, 480, 289, 808, 361, 919, 709, 895, 529, 1591, 750, 1239, 1053, 1711, 841, 2749, 961, 1984, 1681, 2095, 1719, 3660, 1369, 2607, 2323, 3847, 1681, 5091, 1849, 4039, 3673, 3799, 2209, 6519, 2744, 5374

Offset: 1

Author

Franklin T. Adams-Watters, Oct 04 2018

Keywords

• nonn
• look

Comments

sigma(n^2) is always odd, so this sequence has the opposite parity from sigma(n): even if n is a square or twice a square, odd otherwise.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A065764, A072861, A054785.

Programs

Magma

[DivisorSigma(1, n^2) - DivisorSigma(1, n): n in [1..70]]; // Vincenzo Librandi, Oct 05 2018

Maple

map(n -> numtheory:-sigma(n^2)-numtheory:-sigma(n), [$1..100]); # Robert Israel, Oct 04 2018

Mathematica

Table[DivisorSigma[1, n^2] - DivisorSigma[1, n], {n, 70}] (* Vincenzo Librandi, Oct 05 2018 *)

PARI

a(n) = sigma(n^2)-sigma(n)

Python

from _future_ import division
from sympy import factorint
def A320059(n):
    c1, c2 = 1, 1
    for p, a in factorint(n).items():
        c1 *= (p**(2*a+1)-1)//(p-1)
        c2 *= (p**(a+1)-1)//(p-1)
    return c1-c2 # Chai Wah Wu, Oct 05 2018

Formula

a(n) = sigma(n^2) - sigma(n).

a(n) = A065764(n) - A000203(n).

a(n) = n^2 iff n is prime. - Altug Alkan, Oct 04 2018