A382421 The product of exponents in the prime factorization of the noncubefree numbers.

3, 4, 3, 3, 5, 3, 4, 3, 3, 6, 6, 4, 4, 3, 5, 3, 6, 4, 3, 3, 7, 3, 3, 8, 3, 5, 4, 3, 4, 3, 3, 6, 6, 4, 9, 5, 3, 4, 5, 3, 3, 8, 3, 3, 4, 3, 10, 3, 3, 4, 3, 6, 8, 3, 4, 3, 3, 3, 5, 6, 4, 3, 3, 3, 7, 6, 8, 4, 3, 5, 3, 12, 3, 6, 3, 3, 4, 3, 5, 5, 3, 4, 6, 6, 9, 3, 3

Offset: 1

Author

Amiram Eldar, Mar 25 2025

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A002117, A013661, A013664, A005361, A046099, A082695, A330594, A368039, A382419.

Programs

Mathematica

s[n_] := Times @@ FactorInteger[n][[;; , 2]]; noncubeFreeQ[n_] := Max[FactorInteger[n][[;; , 2]]] > 2; s /@ Select[Range[600], noncubeFreeQ]

PARI

list(kmax) = {my(e); for(k = 2, kmax, e = factor(k)[, 2]; if(vecmax(e) > 2, print1(vecprod(e), ", "))); }

Formula

a(n) = A005361(A046099(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (zeta(2)*zeta(3)^2/zeta(6) - zeta(3) * Product_{p prime} (1 + 1/p^2 - 2/p^3))/(zeta(3) - 1) = (A082695 - A330594) * A002117 / (A002117 - 1) = 4.97723390794900554553... .