A073184 Number of cubefree divisors of n.

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

Offset: 1

Author

Reinhard Zumkeller, Jul 19 2002

Keywords

• nonn
• mult

Comments

a(n) = number of divisors of the cubefree kernel of n: a(n) = A000005(A007948(n)); [corrected by Amiram Eldar, Oct 08 2022]

Multiplicative because it is the Inverse Möbius transform of the characteristic function of cubefree numbers. a(n) is a prime signature sequence. a(p) = 2, a(p^e) = 3, e>1. - Christian G. Bower, May 18 2005

Examples

			The divisors of 56 are {1, 2, 4, 7, 8, 14, 28, 56}, 8=2^3 and 56=7*2^3 are not cubefree, therefore a(56) = 6.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000012, A001620, A004709, A027750, A034444, A073180, A073183, A073185, A212793.

Programs

Haskell

a073184 = sum . map a212793 . a027750_row
-- Reinhard Zumkeller, May 27 2012

Mathematica

a[1] = 1; a[p_?PrimeQ] = 2; a[n_] := Times @@ (If[#[[2]] == 1, 2, 3] & /@ FactorInteger[n]); Table[a[n], {n, 1, 103}] (* Jean-François Alcover, May 24 2012, after Christian G. Bower *)

PARI

a(n) = {my(e = factor(n)[,2]); prod(i = 1, #e, if(e[i] == 1, 2, 3))}; \\ Amiram Eldar, Oct 08 2022

Formula

a(n) <= A073182(n).

Dirichlet g.f.: zeta(s)^2/zeta(3*s). Dirichlet convolution of the characteristic function of cubefree numbers by A000012. - R. J. Mathar, Apr 12 2011

a(n) = Sum_{k = 1..A000005(n)} A212793(A027750(n,k)). - Reinhard Zumkeller, May 27 2012

Sum_{k=1..n} a(k) ~ n / zeta(3) * (log(n) - 1 + 2*gamma - 3*zeta'(3)/zeta(3)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 31 2019