A382422 - OEIS

A382422

The product of exponents in the prime factorization of the biquadratefree numbers.

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1

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

OFFSET

1,4

COMMENTS

Differs from A375766 and A375768 at n = 1, 31, 34, 35, 38, 39, ... .

All the terms are 3-smooth numbers (A003586).

LINKS

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

FORMULA

a(n) = A005361(A046100(n)).

a(n) = 2^A382423(n) * 3^A382424(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(4) * Product_{p prime} (1 + 1/p^2 + 1/p^3 - 3/p^4) = 1.57226906210272200398... .

In general, the asymptotic mean of the product of exponents in the prime factorization of the k-free numbers (numbers that are not divisible by a k-th power other than 1), for k >= 2, is zeta(k) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + ... + 1/p^(k-1) - (k-1)/p^k). For k = 2 (squarefree numbers), the mean is 1 since the sequence contains only 1's. The limit when k->oo is zeta(2)*zeta(3)/zeta(6) (A082695).

MATHEMATICA

s[n_] := Times @@ FactorInteger[n][[;; , 2]]; biqFreeQ[n_] := Max[FactorInteger[n][[;; , 2]]] < 4; s /@ Select[Range[100], biqFreeQ]

PROG

(PARI) list(kmax) = {my(e); print1(1, ", "); for(k = 2, kmax, e = factor(k)[, 2]; if(vecmax(e) < 4, print1(vecprod(e), ", "))); }

CROSSREFS

Cf. A003586, A005361, A013662, A046100, A082695, A382423, A382424.

Cf. A375766, A375768.

Sequence in context: A316074 A332954 A115568 * A391898 A375766 A375768

Adjacent sequences: A382419 A382420 A382421 * A382423 A382424 A382425

KEYWORD

nonn,easy

AUTHOR

Amiram Eldar, Mar 25 2025

STATUS

approved