A365488 - OEIS

A365488

The number of divisors of the smallest number whose cube is divisible by n.

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 4, 3, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 2, 4, 2, 8, 2, 3, 4, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 6, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 8, 2, 4, 4, 3, 4, 8, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 4, 8, 2, 6, 3, 4, 2, 8, 4, 4, 4

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

First differs from A365171 at n = 32.

The number of divisors of the smallest cube divisible by n, A053149(n), is A365489(n).

LINKS

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

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

FORMULA

a(n) = A000005(A019555(n)).

Multiplicative with a(p^e) = ceiling(e/3) + 1.

a(n) <= A000005(n) with equality if and only if n is squarefree (A005117).

Dirichlet g.f.: zeta(s) * zeta(3*s) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s)).

From Vaclav Kotesovec, Sep 06 2023: (Start)

Dirichlet g.f.: zeta(s)^2 * zeta(3*s) * Product_{p prime} (1 - 1/p^(2*s) - 1/p^(3*s) + 1/p^(4*s)).

Let f(s) = Product_{p prime} (1 - 1/p^(2*s) - 1/p^(3*s) + 1/p^(4*s)).

Sum_{k=1..n} a(k) ~ zeta(3) * f(1) * n * (log(n) + 2*gamma - 1 + 3*zeta'(3)/zeta(3) + f'(1)/f(1)), where

f(1) = Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.5358961538283379998085026313185459506482223745141452711510108346133288...,

f'(1) = f(1) * Sum_{p prime} (-4 + 3*p + 2*p^2) * log(p) / (1 - p - p^2 + p^4) = f(1) * 1.4525924794451595590371439593828547341482465114411929136723476679...

and gamma is the Euler-Mascheroni constant A001620. (End)

MATHEMATICA

f[p_, e_] := Ceiling[e/3] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

With[{c=Range[200]^3}, Table[DivisorSigma[0, Surd[SelectFirst[c, Mod[#, n]==0&], 3]], {n, 90}]] (* Harvey P. Dale, Sep 15 2024 *)

PROG

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

CROSSREFS

Cf. A000005, A005117, A019555, A053149, A365171, A365489, A365498.

Sequence in context: A034444 A365491 A365210 * A365171 A369310 A318465

Adjacent sequences: A365485 A365486 A365487 * A365489 A365490 A365491

KEYWORD

nonn,easy,mult

AUTHOR

Amiram Eldar, Sep 05 2023

STATUS

approved