A092520 Number of square divisors of n-th cube: a(n) = A046951(n^3).

1, 2, 2, 4, 2, 4, 2, 5, 4, 4, 2, 8, 2, 4, 4, 7, 2, 8, 2, 8, 4, 4, 2, 10, 4, 4, 5, 8, 2, 8, 2, 8, 4, 4, 4, 16, 2, 4, 4, 10, 2, 8, 2, 8, 8, 4, 2, 14, 4, 8, 4, 8, 2, 10, 4, 10, 4, 4, 2, 16, 2, 4, 8, 10, 4, 8, 2, 8, 4, 8, 2, 20, 2, 4, 8, 8, 4, 8, 2, 14, 7, 4, 2, 16, 4, 4, 4, 10, 2, 16, 4, 8, 4, 4, 4, 16

Offset

1,2

Comments

Apparently the inverse Mobius transform of A056624 (and therefore multiplicative). - R. J. Mathar, Feb 07 2011

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Formula

a(n) = A000005(n) iff n is squarefree.

From Werner Schulte, Feb 19 2018: (Start)

Multiplicative with a(p^e) = floor((3*e+2)/2) = A001651(e+1), p prime and e >= 0.

Dirichlet g.f.: Sum_{n>0} a(n)/n^s = (zeta(s))^2 * zeta(2*s) / zeta(3*s). (End)

Sum_{k=1..n} a(k) ~ Pi^2 * n/(6*Zeta(3)) * (log(n) - 1 + 2*gamma + 12*Zeta'(2)/Pi^2 - 3*Zeta'(3)/Zeta(3)) + Zeta(1/2)^2 * sqrt(n) / Zeta(3/2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 08 2019

Example

n=12, the divisors of 12^3 = 1728 are 1=1^2, 2, 3, 4=2^2, 6, 8, 9=3^2, 12, 16=4^2, 18, 24, 27, 32, 36=6^2, 48, 54, 64=8^2, 72, 96, 108, 144=12^2, 192, 216, 288, 432, 576=24^2, 864 and 1728: eight of them are squares, therefore a(12) = 8.

Maple

A092520 := proc(n)
    A046951(n^3) ;
end proc:
seq(A092520(n), n=1..50) ; # R. J. Mathar, Jul 09 2016

Mathematica

a[n_] := Count[Divisors[n^3], d_ /; IntegerQ[Sqrt[d]]]; Array[a, 100] (* Jean-François Alcover, Feb 13 2018 *)
f[p_, e_] := Floor[(3*e+2)/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)

Prog

(PARI) A046951(n) = factorback(apply(e->e\2+1, factor(n)[, 2])) \\ This function from Charles R Greathouse IV, Sep 17 2015
A092520(n) = A046951(n^3); \\ Antti Karttunen, May 25 2017
(PARI) a(n) = sumdiv(n^3, d, issquare(d)); \\ Michel Marcus, Apr 08 2018

Crossrefs

Cf. A000005, A000290, A000578, A005117, A046951, A048785, A056624.

Sequence in context: A118776 A249867 A351612 * A147848 A306652 A239614

Adjacent sequences: A092517 A092518 A092519 * A092521 A092522 A092523

Keyword

nonn,mult

Author

Reinhard Zumkeller, Apr 06 2004

Status

approved