A382419 The product of exponents in the prime factorization of the cubefree numbers.

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 4, 1, 1

Offset: 1

Author

Amiram Eldar, Mar 25 2025

Keywords

• nonn
• easy

Comments

Differs from A368712 at n = 1, 31, 85, 151, 164, 189, ... .

All the terms are powers of 2.

Links

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

Crossrefs

Cf. A002117, A004709, A005361, A330594, A368712, A376366, A382421, A382422.

Programs

Mathematica

s[n_] := Times @@ FactorInteger[n][[;; , 2]]; cubeFreeQ[n_] := Max[FactorInteger[n][[;; , 2]]] < 3; s /@ Select[Range[120], cubeFreeQ]

PARI

list(kmax) = {my(e); print1(1, ", "); for(k = 2, kmax, e = factor(k)[, 2]; if(vecmax(e) < 3, print1(vecprod(e), ", "))); }

Formula

a(n) = A005361(A004709(n)).

a(n) = 2^A376366(n).

a(n) >= A368712(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3) * Product_{p prime} (1 + 1/p^2 - 2/p^3) = A002117 * A330594 = 1.33062904409568262931... .