A376365 The number of distinct prime factors of the cubefree numbers.

0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 3, 1, 2, 2, 2, 2, 1, 2, 2, 1, 3, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 3, 1, 2, 2, 3, 1, 1, 2, 2, 2, 2, 3, 1, 2, 1, 3, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 3, 2, 1, 1, 3, 2, 1, 3, 2, 2, 2, 2, 2, 1, 2, 2, 2

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, Distribution of omega(n) over h-free and h-full numbers, arXiv:2409.10430 [math.NT], 2024. See Theorem 1.1.

Crossrefs

Cf. A001221, A004709, A059956, A072047, A077761, A376361, A376363, A376366.

Programs

Mathematica

f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # < 3 &], Length[e], 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(#e, ", ")));}

Formula

a(n) = A001221(A004709(n)).

Sum_{A004709(k) <= x} a(k) = (6/Pi^2) * x * (log(log(x)) + B - C) + O(x/log(x)), where B is Mertens's constant (A077761) and C = Sum_{p prime} (p-1)/(p*(p^3-1)) = 0.10770743252352371604... (Das et al., 2024).