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

%I A376366 #25 Apr 05 2025 09:10:54
%S A376366 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,
%T A376366 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,
%U A376366 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
%N A376366 The number of non-unitary prime divisors of the cubefree numbers.
%H A376366 Amiram Eldar, <a href="/A376366/b376366.txt">Table of n, a(n) for n = 1..10000</a>
%H A376366 Sourabhashis Das, Wentang Kuo, and Yu-Ru Liu, <a href="https://doi.org/10.1016/j.jnt.2024.08.007">On the number of prime factors with a given multiplicity over h-free and h-full numbers</a>, Journal of Number Theory, Vol. 267 (2025), pp. 176-201; <a href="https://arxiv.org/abs/2409.11275">arXiv preprint</a>, arXiv:2409.11275 [math.NT], 2024. See Theorem 1.2.
%F A376366 a(n) = A056170(A004709(n)).
%F A376366 a(n) = A369427(A004709(n)).
%F A376366 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).
%F A376366 a(n) = log_2(A382419(n)). - _Amiram Eldar_, Mar 25 2025
%F A376366 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)
%F A376366 From _Amiram Eldar_, Apr 05 2025: (Start)
%F A376366 a(n) = A046660(A004709(n)).
%F A376366 a(n) = A368779(n) - A376365(n). (End)
%t A376366 f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # < 3 &], Count[e, 2], Nothing]]; Array[f, 150]
%o A376366 (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), ", ")));}
%Y A376366 Cf. A002117, A004709, A046660, A056170, A088453, A368779, A369427, A376362, A376365, A382419.
%K A376366 nonn,easy
%O A376366 1,31
%A A376366 _Amiram Eldar_, Sep 21 2024