A382419 - OEIS

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

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OFFSET

1,4

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

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... .

From Amiram Eldar, Dec 14 2025: (Start)

a(n) = A049419(A004709(n)).

a(n) = A278908(A004709(n)). (End)

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A002117, A004709, A005361, A049419, A278908, A330594, A368712, A376366, A382421, A382422.

Sequence in context: A198254 A309059 A368712 * A293630 A305393 A392014

Adjacent sequences: A382416 A382417 A382418 * A382420 A382421 A382422

KEYWORD

nonn,easy

AUTHOR

Amiram Eldar, Mar 25 2025

STATUS

approved