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

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, 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, 2, 3, 1, 2, 1, 2, 3, 3, 2, 3, 1, 1, 0, 2, 1, 4, 2, 2, 2

Offset: 1

Author

Amiram Eldar, Sep 28 2023

Keywords

• nonn
• easy

Comments

The sum of exponents smaller than 3 in the prime factorization of n.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Rafael Jakimczuk and Matilde Lalín, Sums of omega(n) and Omega(n) over the k-free parts and k-full parts of some particular sequences, Integers, Vol. 22 (2022), Article #A113.

Crossrefs

Cf. A001222, A004709, A005117, A036966, A056169, A077761, A085541, A085548, A360539, A366076, A366078.

Programs

Mathematica

f[p_, e_] := If[e < 3, e, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = vecsum(apply(x -> if(x < 3, x, 0), factor(n)[, 2]));

Formula

a(n) = A001222(A360539(n)).

a(n) = A001222(n) - A366076(n).

Additive with a(p^e) = e if e <= 2, and a(p^e) = 0 for e >= 3.

a(n) >= 0, with equality if and only if n is cubefull (A036966).

a(n) <= A001222(n), with equality if and only if n is cubefree (A004709).

a(n) >= A366078(n), with equality if and only if n is squarefree (A005117).

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... .