A366440 The sum of divisors of the cubefree numbers (A004709).

1, 3, 4, 7, 6, 12, 8, 13, 18, 12, 28, 14, 24, 24, 18, 39, 20, 42, 32, 36, 24, 31, 42, 56, 30, 72, 32, 48, 54, 48, 91, 38, 60, 56, 42, 96, 44, 84, 78, 72, 48, 57, 93, 72, 98, 54, 72, 80, 90, 60, 168, 62, 96, 104, 84, 144, 68, 126, 96, 144, 72, 74, 114, 124, 140

Offset

1,2

Links

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

Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634; p. 619, eq. (2).

Formula

a(n) = A000203(A004709(n)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = 15*zeta(3)/(2*Pi^2) = A082020 * A002117 / 2 = 0.913453711751... .

The asymptotic mean of the abundancy index of the cubefree numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A004709(k) = 15/Pi^2 = 1.519817... (A082020).

Mathematica

f[p_, e_] := (p^(e+1)-1)/(p-1); s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 3 &], Times @@ f @@@ fct, Nothing]]; s[1] = 1; Array[s, 100]

Prog

(PARI) lista(max) = for(k = 1, max, my(f = factor(k), iscubefree = 1); for(i = 1, #f~, if(f[i, 2] > 2, iscubefree = 0; break)); if(iscubefree, print1(sigma(f), ", ")));
(Python)
from sympy import mobius, integer_nthroot, divisor_sigma
def A366440(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 divisor_sigma(m) # Chai Wah Wu, Aug 06 2024

Crossrefs

Cf. A000203, A004709, A358040.

Cf. A002117, A082020.

Similar sequences: A062822, A065764, A180114, A362986, A366439.

Sequence in context: A049418 A333926 A051378 * A344575 A254981 A116607

Adjacent sequences: A366437 A366438 A366439 * A366441 A366442 A366443

Keyword

nonn,easy

Author

Amiram Eldar, Oct 10 2023

Status

approved