A368248 The number of unitary divisors of the cubefull part of n (A360540).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Dec 19 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A061704 and A362852 at n = 64, and from A304327 at n = 72.

Also, the number of squarefree divisors of the cubefull part of n.

Also, the number of cubes of squarefree numbers (A062838) that divide n.

The number of unitary divisors of n that are cubefull numbers (A036966). - Amiram Eldar, Jun 19 2025

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A004709, A034444, A036966, A062838, A157289, A307428, A323308, A360540, A366145.

Cf. A061704, A304327, A362852.

Programs

Mathematica

f[p_, e_] := If[e > 2, 2, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = vecprod(apply(x -> if(x < 3, 1, 2), factor(n)[, 2]));

Formula

a(n) = A034444(A360540(n)).

a(n) = abs(A307428(n)).

Multiplicative with a(p) = 1 for e <= 2, and 2 for e >= 3.

a(n) >= 1, with equality if and only if n is cubefree (A004709).

a(n) <= A034444(n), with equality if and only if n is cubefull (A036966).

Dirichlet g.f.: zeta(s)*zeta(3*s)/zeta(6*s).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3)/zeta(6) = 1.181564... (A157289).

In general, the asymptotic mean of the number of unitary divisors of the k-full part of n is zeta(k)/zeta(2*k).