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

1,8

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.

Links

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

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).

Mathematica

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

Prog

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

Crossrefs

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

Cf. A061704, A304327, A362852.

Sequence in context: A320267 A304327 A307428 * A362852 A061704 A325837

Adjacent sequences: A368245 A368246 A368247 * A368249 A368250 A368251

Keyword

nonn,easy,mult

Author

Amiram Eldar, Dec 19 2023

Status

approved