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A363013 a(n) is the number of prime factors (counted with multiplicity) of the n-th cubefull number (A036966).

%I A363013 #13 Sep 21 2024 14:46:12
%S A363013 0,3,4,3,5,6,4,3,7,6,5,8,3,7,9,4,7,6,8,6,10,8,3,9,8,7,11,7,3,4,9,6,5,
%T A363013 6,10,9,8,12,3,7,10,7,9,8,3,11,10,9,13,6,8,7,11,6,8,10,3,12,4,11,6,10,
%U A363013 14,5,7,10,6,7,9,9,12,7,9,11,3,8,9,13,7,4,3
%N A363013 a(n) is the number of prime factors (counted with multiplicity) of the n-th cubefull number (A036966).
%H A363013 Amiram Eldar, <a href="/A363013/b363013.txt">Table of n, a(n) for n = 1..10000</a>
%H A363013 Rafael Jakimczuk and Matilde Lalín, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Lalin/lalin2.html">The Number of Prime Factors on Average in Certain Integer Sequences</a>, Journal of Integer Sequences, Vol. 25 (2022), Article 22.2.3.
%F A363013 a(n) = A001222(A036966(n)).
%F A363013 a(n) >= 3, for n > 1.
%F A363013 Sum_{A036966(k) < x} a(k) = 3*c*x^(1/3)*log(log(x)) + (3*(B_2 - log(2)) + Sum_{p prime} ((4*p^(1/3)+5)/(p^(5/3)+p^(1/3)+1)))*c*x^(1/3) + O(x^(1/3)/sqrt(log(x))), where B_2 = A083342 and c = A362974 (Jakimczuk and Lalín, 2022). [corrected Sep 21 2024]
%t A363013 PrimeOmega[Select[Range[10000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 2 &]]
%o A363013 (PARI) iscubefull(n) = n==1 || vecmin(factor(n)[, 2]) > 2;
%o A363013 apply(bigomega, select(iscubefull, [1..10000]))
%Y A363013 Cf. A001222, A036966, A362974.
%Y A363013 Similar sequences: A072047, A076399, A360729.
%K A363013 nonn
%O A363013 1,2
%A A363013 _Amiram Eldar_, May 13 2023