A369889 - OEIS

A369889

The sum of squarefree divisors of the cubefree numbers.

1, 3, 4, 3, 6, 12, 8, 4, 18, 12, 12, 14, 24, 24, 18, 12, 20, 18, 32, 36, 24, 6, 42, 24, 30, 72, 32, 48, 54, 48, 12, 38, 60, 56, 42, 96, 44, 36, 24, 72, 48, 8, 18, 72, 42, 54, 72, 80, 90, 60, 72, 62, 96, 32, 84, 144, 68, 54, 96, 144, 72, 74, 114, 24, 60, 96, 168

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

OFFSET

1,2

COMMENTS

The number of squarefree divisors of the n-th cubefree number is A366536(n).

LINKS

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

FORMULA

a(n) = A048250(A004709(n)).

Sum_{j=1..n} a(j) ~ c * n^2, where c = zeta(3)^2/(2*zeta(5)) = 0.6967413068... .

In general, the formula holds for the sum of squarefree divisors of the k-free numbers with c = zeta(k)^2/(2*zeta(2*k-1))..., for k >= 2.

MATHEMATICA

f[p_, e_] := p + 1; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; cubefreeQ[n_] := Max[FactorInteger[n][[;; , 2]]] < 3; s /@ Select[Range[100], cubefreeQ]

(* or *)

f[p_, e_] := If[e > 2, 0, p + 1]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 100], # > 0 &]

PROG

(PARI) lista(kmax) = {my(f, s, p, e); for(k = 1, kmax, f = factor(k); s = prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; if(e < 3, p + 1, 0)); if(s > 0, print1(s, ", "))); }

(Python)

from math import prod

from sympy import mobius, integer_nthroot, primefactors

def A369889(n):

def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x, 3)[0]+1))

m, k = n, f(n)

while m != k:

m, k = k, f(k)

return prod(p+1 for p in primefactors(m)) # Chai Wah Wu, Aug 12 2024

CROSSREFS

Cf. A004709, A048250, A062822, A366440, A366536, A366537.

Cf. A002117, A013663.