A369306 The number of cubefree divisors d of n such that n/d is also cubefree.

1, 2, 2, 3, 2, 4, 2, 2, 3, 4, 2, 6, 2, 4, 4, 1, 2, 6, 2, 6, 4, 4, 2, 4, 3, 4, 2, 6, 2, 8, 2, 0, 4, 4, 4, 9, 2, 4, 4, 4, 2, 8, 2, 6, 6, 4, 2, 2, 3, 6, 4, 6, 2, 4, 4, 4, 4, 4, 2, 12, 2, 4, 6, 0, 4, 8, 2, 6, 4, 8, 2, 6, 2, 4, 6, 6, 4, 8, 2, 2, 1, 4, 2, 12, 4, 4, 4

Offset: 1

Author

Amiram Eldar, Jan 19 2024

Keywords

• nonn
• easy
• mult

Comments

The analogous sequence with squarefree divisors (the number of squarefree divisors d of n such that n/d is also squarefree) is abs(A007427(n)).

Links

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

Crossrefs

Cf. A000005, A004709, A005117, A007427, A069492, A073184.

Cf. A001620, A002117, A244115.

Programs

Mathematica

f[p_,e_] := Switch[e, 1, 2, 2, 3, 3, 2, 4, 1, , 0]; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = vecprod(apply(x -> [2, 3, 2, 1, 0][min(x, 5)], factor(n)[,2]));

Formula

Multiplicative with a(p) = 2, a(p^2) = 3, a(p^3) = 2, a(p^4) = 1, and a(p^e) = 0 for e >= 5.

a(n) >= 0, with equality if and only if n is a 5-full number (A069492) larger than 1.

a(n) = 1 if and only if n is the 4th power of a squarefree number (A005117).

a(n) <= A000005(n), with equality if and only if n is cubefree (A004709).

Dirichlet g.f.: zeta(s)^2/zeta(3*s)^2.

Sum_{k=1..n} a(k) ~ (n/zeta(3)^2) * (log(n) + 2*gamma - 1 - 6*zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620).