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A376363 The number of distinct prime factors of the cubefull numbers.

%I A376363 #13 Mar 25 2025 01:27:44
%S A376363 0,1,1,1,1,1,1,1,1,2,1,1,1,2,1,1,2,1,2,2,1,2,1,2,2,2,1,1,1,1,2,2,1,2,
%T A376363 2,2,2,1,1,2,2,2,2,1,1,2,2,2,1,2,2,2,2,2,2,2,1,2,1,2,1,2,1,1,2,2,2,2,
%U A376363 1,2,2,2,2,2,1,2,3,2,2,1,1,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,3,2,2,1,2
%N A376363 The number of distinct prime factors of the cubefull numbers.
%H A376363 Amiram Eldar, <a href="/A376363/b376363.txt">Table of n, a(n) for n = 1..10000</a>
%H A376363 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 1.2.
%H A376363 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%F A376363 a(n) = A001221(A036966(n)).
%F A376363 Sum_{A036966(k) <= x} a(k) = c * x^(1/3) * (log(log(x)) + B - log(3) + L(3, 4) - L(3, 6)) + O(x^(1/3)/log(x)), where c = A362974, B is Mertens's constant (A077761), L(h, r) = Sum_{p prime} 1/(p^(r/h - 1) * (p - p^(1 - 1/h) + 1)), L(3, 4) = 1.65235055631578303808..., and L(3, 6) = 0.67060646664392140547... (Das et al., 2024).
%t A376363 f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # > 2 &], Length[e], Nothing]]; Array[f, 60000]
%o A376363 (PARI) lista(kmax) = {my(e, is); for(k = 1, kmax, e = factor(k)[, 2]; is = 1; for(i = 1, #e, if(e[i] < 3, is = 0; break)); if(is, print1(#e, ", ")));}
%Y A376363 Cf. A001221, A036966, A072047, A077761, A362974, A376361, A376364, A376365.
%K A376363 nonn,easy
%O A376363 1,10
%A A376363 _Amiram Eldar_, Sep 21 2024