A373058 The sum of the aliquot coreful divisors of the nonsquarefree numbers.

2, 6, 3, 6, 14, 6, 10, 18, 5, 12, 14, 30, 36, 30, 22, 15, 42, 7, 10, 26, 24, 42, 30, 21, 62, 34, 96, 15, 38, 70, 39, 42, 66, 30, 46, 90, 14, 33, 80, 78, 126, 98, 58, 39, 90, 11, 62, 30, 42, 126, 66, 60, 102, 70, 216, 21, 74, 30, 114, 51, 78, 150, 78, 82, 126, 13

Offset: 1

Author

Amiram Eldar, May 21 2024

Keywords

• nonn
• easy

Comments

A coreful divisor d of n is a divisor that is divisible by every prime that divides n (see also A307958).

The positive terms of A336563: if k is a squarefree number (A005117) then the only coreful divisor of k is k itself, so k has no aliquot coreful divisors.

The number of the aliquot coreful divisors of the n-th nonsquarefree number is A368039(n).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005117, A013661, A013929, A065487, A229099, A307958, A336563.

Cf. A084936, A174961, A275699, A368038, A368039, A368040, A368541, A368713.

Programs

Mathematica

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; Select[Array[s, 300], # > 0 &]

PARI

lista(kmax) = {my(f); for(k = 1, kmax, f = factor(k); if(!issquarefree(f), print1(prod(i = 1, #f~, (f[i, 1]^(f[i, 2]+1) - 1)/(f[i, 1] - 1) - 1) - k, ", "))); }

Python

from math import prod, isqrt
from sympy import mobius, factorint
def A373058(n):
    def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return prod((p**(e+1)-1)//(p-1)-1 for p, e in factorint(m).items())-m # Chai Wah Wu, Jul 22 2024

Formula

a(n) = A336563(A013929(n)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (A065487 - 1)/(1-1/zeta(2))^2 = 1.50461493205911656114... .