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A073184 Number of cubefree divisors of n. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) = sum of divisors of the cubefree kernel of n: a(n) = A073184(A007948(n));

a(n) <= A073182(n).

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

LINKS

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

FORMULA

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(A212793(A027750(n,k)): k = 1..A000005(n)). - Reinhard Zumkeller, May 27 2012

EXAMPLE

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.

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 *)

PROG

(Haskell)

a073184 = sum . map a212793 . a027750_row

-- Reinhard Zumkeller, May 27 2012

CROSSREFS

Cf. A000005, A073185, A004709, A073183, A073180, A034444.

Sequence in context: A217984 A196437 A106491 * A073182 A282446 A049599

Adjacent sequences:  A073181 A073182 A073183 * A073185 A073186 A073187

KEYWORD

nonn,mult

AUTHOR

Reinhard Zumkeller, Jul 19 2002

STATUS

approved

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Last modified April 30 08:33 EDT 2017. Contains 285645 sequences.