A380395 The number of unitary divisors of n that are cubes.

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

Offset: 1

Author

Amiram Eldar, Jan 23 2025

Keywords

• nonn
• easy
• mult

Comments

First differs from A318672 at n = 64.

The sum of unitary divisors of n that are cubes is A380396(n).

Examples

			a(8) = 2 since 8 has 2 unitary divisors that are cubes, 1 = 1^3 and 8 = 2^3.
a(216) = 4 since 216 has 4 unitary divisors that are cubes, 1 = 1^3, 8 = 2^3, 27 = 3^3 and 216 = 6^3.
		

Links

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

Crossrefs

Cf. A000578, A046100, A056624, A061704, A077610, A307427, A318672, A366761, A380396, A380397.

Cf. A002117, A013662.

Programs

Mathematica

f[p_, e_] := 2^If[Divisible[e, 3], 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 2^if(f[i, 2]%3, 0, 1));}

Formula

a(n) = Sum_{d|n, gcd(d, n/d) = 1} [d is cube], where [] is the Iverson bracket.

Multiplicative with a(p^e) = 2 is e is divisible by 3, and 1 otherwise.

a(n) = abs(A307427(n)).

a(n) = A061704(n) - A380397(n).

a(n) >= 1, with equality if and only if n is not in A366761.

a(n) <= A061704(n), with equality if and only if n is biquadratefree (A046100).

Dirichlet g.f.: zeta(s)*zeta(3*s)/zeta(4*s).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3)/zeta(4) = 1.11062653532614811717... .

In general, the asymptotic mean of the number of unitary divisors of n that are m-th powers is zeta(m)/zeta(m+1), for m >= 2.