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A366076 The number of prime factors of the largest divisor of n that is a cubefull number (A036966), counted with multiplicity.

%I A366076 #12 May 07 2025 10:45:41
%S A366076 0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,3,0,0,0,0,5,0,0,
%T A366076 0,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,6,0,0,0,0,
%U A366076 0,0,0,3,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0
%N A366076 The number of prime factors of the largest divisor of n that is a cubefull number (A036966), counted with multiplicity.
%C A366076 The sum of exponents larger than 2 in the prime factorization of n.
%C A366076 The number of distinct prime factors of the largest divisor of n that is a cubefull number is A295659(n).
%H A366076 Amiram Eldar, <a href="/A366076/b366076.txt">Table of n, a(n) for n = 1..10000</a>
%H A366076 Rafael Jakimczuk and Matilde Lalín, <a href="http://math.colgate.edu/~integers/w113/w113.pdf">Sums of omega(n) and Omega(n) over the k-free parts and k-full parts of some particular sequences</a>, Integers, Vol. 22 (2022), Article #A113.
%F A366076 a(n) = A001222(A360540(n)).
%F A366076 a(n) = A001222(n) - A366077(n).
%F A366076 Additive with a(p^e) = 0 if e <= 2, and a(p^e) = e for e >= 3.
%F A366076 a(n) >= 0, with equality if and only if n is cubefree (A004709).
%F A366076 a(n) <= A001222(n), with equality if and only if n is cubefull (A036966).
%F A366076 a(n) >= 3*A295659(n), with equality if and only if n is a biquadratefree number (A046100).
%F A366076 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (2/p^3 + 1/(p^2*(p-1))) = 2 * A085541 + A152441 = 0.67043452760761670220... .
%t A366076 f[p_, e_] := If[e < 3, 0, e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A366076 (PARI) a(n) = vecsum(apply(x -> if(x < 3, 0, x), factor(n)[, 2]));
%Y A366076 Cf. A001222, A004709, A036966, A046100, A085541, A152441, A295659, A360540, A366077.
%Y A366076 Similar sequence: A275812 (number of prime factors of the powerful part).
%K A366076 nonn,easy
%O A366076 1,8
%A A366076 _Amiram Eldar_, Sep 28 2023