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

1,4

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.

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... .

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A118832 A122807 A105700 * A367169 A236831 A030205

Adjacent sequences: A366074 A366075 A366076 * A366078 A366079 A366080

Keyword

nonn,easy

Author

Amiram Eldar, Sep 28 2023

Status

approved