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A366077 The number of prime factors of the cubefree part of n (A360539), counted with multiplicity.

%I A366077 #9 Sep 29 2023 04:10:36
%S A366077 0,1,1,2,1,2,1,0,2,2,1,3,1,2,2,0,1,3,1,3,2,2,1,1,2,2,0,3,1,3,1,0,2,2,
%T A366077 2,4,1,2,2,1,1,3,1,3,3,2,1,1,2,3,2,3,1,1,2,1,2,2,1,4,1,2,3,0,2,3,1,3,
%U A366077 2,3,1,2,1,2,3,3,2,3,1,1,0,2,1,4,2,2,2
%N A366077 The number of prime factors of the cubefree part of n (A360539), counted with multiplicity.
%C A366077 The sum of exponents smaller than 3 in the prime factorization of n.
%H A366077 Amiram Eldar, <a href="/A366077/b366077.txt">Table of n, a(n) for n = 1..10000</a>
%H A366077 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 A366077 a(n) = A001222(A360539(n)).
%F A366077 a(n) = A001222(n) - A366076(n).
%F A366077 Additive with a(p^e) = e if e <= 2, and a(p^e) = 0 for e >= 3.
%F A366077 a(n) >= 0, with equality if and only if n is cubefull (A036966).
%F A366077 a(n) <= A001222(n), with equality if and only if n is cubefree (A004709).
%F A366077 a(n) >= A366078(n), with equality if and only if n is squarefree (A005117).
%F A366077 Sum_{k=1..m} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} (p-2)/p^3 = A085548 - 2 * A085541 = 0.10272214144217842566... .
%t A366077 f[p_, e_] := If[e < 3, e, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A366077 (PARI) a(n) = vecsum(apply(x -> if(x < 3, x, 0), factor(n)[, 2]));
%Y A366077 Cf. A001222, A004709, A005117, A036966, A056169, A077761, A085541, A085548, A360539, A366076, A366078.
%K A366077 nonn,easy
%O A366077 1,4
%A A366077 _Amiram Eldar_, Sep 28 2023