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

1,2

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

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).

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]

Prog

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

Crossrefs

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

Cf. A001620, A002117, A244115.

Sequence in context: A359302 A366147 A367170 * A038148 A366742 A141829

Adjacent sequences: A369303 A369304 A369305 * A369307 A369308 A369309

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 19 2024

Status

approved