A368039 The product of exponents of prime factorization of the nonsquarefree numbers.

2, 3, 2, 2, 4, 2, 2, 3, 2, 3, 2, 5, 4, 3, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 6, 2, 6, 2, 2, 4, 4, 2, 3, 2, 2, 5, 2, 2, 4, 3, 6, 4, 2, 2, 3, 2, 2, 3, 2, 7, 2, 3, 3, 2, 8, 2, 2, 2, 3, 2, 2, 5, 4, 2, 3, 2, 2, 2, 2, 4, 4, 3, 2, 3, 6, 4, 2, 6, 2, 2, 4, 2, 9, 2, 5, 4, 2

Offset: 1

Author

Amiram Eldar, Dec 09 2023

Keywords

• nonn

Comments

The terms of A005361 that are larger than 1, since A005361(k) = 1 if and only if k is squarefree (A005117).

Links

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

Crossrefs

Cf. A005117, A005361, A013929.

Cf. A084936, A174961, A275699, A368038, A368040.

Cf. A059956, A082695, A229099.

Programs

Mathematica

Select[Table[Times @@ FactorInteger[n][[;;, 2]], {n, 1, 250}], # > 1 &]

PARI

lista(kmax) = {my(p); for(k = 1, kmax, p = vecprod(factor(k)[, 2]); if(p > 1, print1(p, ", ")));}

Formula

a(n) = A005361(A013929(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = ((zeta(2)*zeta(3)/zeta(6)) - 1/zeta(2))/(1-1/zeta(2)) = (A082695 - A059956)/A229099 = 3.406686208821... .