A382965 The number of non-unitary prime divisors of the n-th cubefree number that is not squarefree.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Apr 10 2025

Keywords

• nonn
• easy

Comments

The positive terms of A376366.

Links

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

Crossrefs

Cf. A002117, A046660, A056170, A067259, A013661, A369427, A376366, A382966, A382968.

Programs

Mathematica

f[k_] := If[k == 1, Nothing, Module[{e = FactorInteger[k][[;; , 2]]}, If[Max[e] == 2, Count[e, 2], Nothing]]]; Array[f, 150]

PARI

list(lim) = {my(e); for(k = 2, lim, e = factor(k)[, 2]; if(vecmax(e) == 2, print1(#select(x -> x == 2, e), ", ")));}

Formula

a(n) = A056170(A067259(n)).

a(n) = A046660(A067259(n)).

a(n) = A369427(A067259(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (zeta(2)/(zeta(2)-zeta(3))) * Sum_{p prime} ((p-1)/(p^3-1)) = 1.10872412837037270926... .