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

%I A376365 #13 Mar 25 2025 01:28:05
%S A376365 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,
%T A376365 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,
%U A376365 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
%N A376365 The number of distinct prime factors of the cubefree numbers.
%H A376365 Amiram Eldar, <a href="/A376365/b376365.txt">Table of n, a(n) for n = 1..10000</a>
%H A376365 Sourabhashis Das, Wentang Kuo, and Yu-Ru Liu, <a href="https://arxiv.org/abs/2409.10430">Distribution of omega(n) over h-free and h-full numbers</a>, arXiv:2409.10430 [math.NT], 2024. See Theorem 1.1.
%F A376365 a(n) = A001221(A004709(n)).
%F A376365 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).
%t A376365 f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # < 3 &], Length[e], Nothing]]; Array[f, 150]
%o A376365 (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, ", ")));}
%Y A376365 Cf. A001221, A004709, A059956, A072047, A077761, A376361, A376363, A376366.
%K A376365 nonn,easy
%O A376365 1,6
%A A376365 _Amiram Eldar_, Sep 21 2024