A366076 The number of primes 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

1,8

Comments

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

The number of distinct primes 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.

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... .

Mathematica

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

Prog

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

Crossrefs

Cf. A001222, A004709, A036966, A046100, A085541, A152441, A295659, A360540, A366077.

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

Sequence in context: A318673 A151795 A008437 * A324326 A341743 A348945

Adjacent sequences: A366073 A366074 A366075 * A366077 A366078 A366079

Keyword

nonn,easy

Author

Amiram Eldar, Sep 28 2023

Status

approved