A073184 - OEIS

%I #36 Nov 13 2022 05:30:13

%S 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,

%T 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,

%U 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

%N Number of cubefree divisors of n.

%C a(n) = number of divisors of the cubefree kernel of n: a(n) = A000005(A007948(n)); [corrected by _Amiram Eldar_, Oct 08 2022]

%C 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

%H Reinhard Zumkeller, <a href="/A073184/b073184.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) <= A073182(n).

%F 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

%F a(n) = Sum_{k = 1..A000005(n)} A212793(A027750(n,k)). - _Reinhard Zumkeller_, May 27 2012

%F 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

%e 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.

%t 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_ *)

%o (Haskell)

%o a073184 = sum . map a212793 . a027750_row

%o -- _Reinhard Zumkeller_, May 27 2012

%o (PARI) a(n) = {my(e = factor(n)[,2]); prod(i = 1, #e, if(e[i] == 1, 2, 3))}; \\ _Amiram Eldar_, Oct 08 2022

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

%K nonn,mult

%O 1,2

%A _Reinhard Zumkeller_, Jul 19 2002