A376366 The number of non-unitary prime divisors of the cubefree numbers.

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1

Offset: 1

Author

Amiram Eldar, Sep 21 2024

Keywords

• nonn
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Sourabhashis Das, Wentang Kuo, and Yu-Ru Liu, On the number of prime factors with a given multiplicity over h-free and h-full numbers, Journal of Number Theory, Vol. 267 (2025), pp. 176-201; arXiv preprint, arXiv:2409.11275 [math.NT], 2024. See Theorem 1.2.

Crossrefs

Cf. A002117, A004709, A046660, A056170, A088453, A368779, A369427, A376362, A376365, A382419.

Programs

Mathematica

f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # < 3 &], Count[e, 2], Nothing]]; Array[f, 150]

PARI

lista(kmax) = {my(e, is); for(k = 1, kmax, e = factor(k)[, 2]; is = 1; for(i = 1, #e, if(e[i] > 2, is = 0; break)); if(is, print1(#select(x -> x == 2, e), ", ")));}

Formula

a(n) = A056170(A004709(n)).

a(n) = A369427(A004709(n)).

Sum_{A004709(k) <= x} a(k) = c * x + O(sqrt(x)/log(x)), where c = (1/zeta(3)) * Sum_{p prime} ((p-1)/(p^3-1)) = 0.24833233043359932037... (Das et al., 2025).

a(n) = log_2(A382419(n)). - Amiram Eldar, Mar 25 2025

Sum_{k=1..n} a(k) ~ c * n, where c = Sum_{p prime} ((p-1)/(p^3-1)) = 0.29850959207541746... - Vaclav Kotesovec, Mar 25 2025 (according to the above formula)

From Amiram Eldar, Apr 05 2025: (Start)

a(n) = A046660(A004709(n)).

a(n) = A368779(n) - A376365(n). (End)