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

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, 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, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Amiram Eldar, Sep 28 2023

Keywords

• nonn
• easy

Comments

The sum of exponents larger than 2 in the prime factorization of n.

The number of distinct prime factors of the largest divisor of n that is a cubefull number is A295659(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, A036966, A046100, A085541, A152441, A295659, A360540, A366077.

Similar sequence: A275812 (number of prime factors of the powerful part).

Programs

Mathematica

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

PARI

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

Formula

a(n) = A001222(A360540(n)).

a(n) = A001222(n) - A366077(n).

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

a(n) >= 0, with equality if and only if n is cubefree (A004709).

a(n) <= A001222(n), with equality if and only if n is cubefull (A036966).

a(n) >= 3*A295659(n), with equality if and only if n is a biquadratefree number (A046100).

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... .