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A366078 The number of distinct prime factors of the cubefree part of n (A360539).

%I A366078 #7 Sep 29 2023 04:06:39
%S A366078 0,1,1,1,1,2,1,0,1,2,1,2,1,2,2,0,1,2,1,2,2,2,1,1,1,2,0,2,1,3,1,0,2,2,
%T A366078 2,2,1,2,2,1,1,3,1,2,2,2,1,1,1,2,2,2,1,1,2,1,2,2,1,3,1,2,2,0,2,3,1,2,
%U A366078 2,3,1,1,1,2,2,2,2,3,1,1,0,2,1,3,2,2,2
%N A366078 The number of distinct prime factors of the cubefree part of n (A360539).
%C A366078 The number of exponents smaller than 3 in the prime factorization of n.
%C A366078 The number of prime factors of the cubefree part of n (A360539), counted with multiplicity is A366077(n).
%H A366078 Amiram Eldar, <a href="/A366078/b366078.txt">Table of n, a(n) for n = 1..10000</a>
%H A366078 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 A366078 a(n) = A001221(A360539(n)).
%F A366078 a(n) = A001221(n) - A295659(n).
%F A366078 Additive with a(p^e) = 1 if e <= 2, and a(p^e) = 0 for e >= 3.
%F A366078 a(n) >= 0, with equality if and only if n is cubefull (A036966).
%F A366078 a(n) <= A001221(n), with equality if and only if n is cubefree (A004709).
%F A366078 a(n) <= A366077(n), with equality if and only if n is squarefree (A005117).
%F A366078 Sum_{k=1..m} a(k) = n * (log(log(n)) + B - C) + O(n/log(n)), where B is Mertens's constant (A077761) and C = Sum_{p prime} 1/p^3 = 0.174762... (A085541).
%t A366078 f[p_, e_] := If[e < 3, 1, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A366078 (PARI) a(n) = vecsum(apply(x -> if(x < 3, 1, 0), factor(n)[, 2]));
%Y A366078 Cf. A001221, A004709, A005117, A036966, A077761, A085541, A295659, A360539, A366077.
%K A366078 nonn,easy
%O A366078 1,6
%A A366078 _Amiram Eldar_, Sep 28 2023