The On-Line Encyclopedia of Integer Sequences (OEIS)

A373058 revision #13

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

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... .

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 &]

PROG

(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

CROSSREFS

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

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