A050326 Number of factorizations of n into distinct squarefree numbers > 1.

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 5, 1, 0, 2, 2, 2, 1, 1, 2, 2, 0, 1, 5, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 4, 1, 2, 1, 0, 2, 5, 1, 1, 2, 5, 1, 0, 1, 2, 1, 1, 2, 5, 1, 0, 0, 2, 1, 4, 2, 2, 2, 0, 1, 4, 2, 1, 2, 2, 2, 0, 1, 1, 1, 1, 1, 5, 1

Offset: 1

Author

Christian G. Bower, Oct 15 1999

Keywords

• nonn

Comments

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1).

a(A212164(n)) = 0; a(A212166(n)) = 1; a(A006881(n)) = 2; a(A190107(n)) = 3; a(A085987(n)) = 4; a(A225228(n)) = 5; a(A179670(n)) = 7; a(A162143(n)) = 8; a(A190108(n)) = 11; a(A212167(n)) > 0; a(A212168(n)) > 1. - Reinhard Zumkeller, May 03 2013

The comment that a(A212164(n)) = 0 is incorrect. For example, 3600 belongs to A212164 but a(3600) = 1. The positions of zeros in this sequence are A293243. - Gus Wiseman, Oct 10 2017

Examples

			The a(30) = 5 factorizations are: 2*3*5, 2*15, 3*10, 5*6, 30. The a(180) = 5 factorizations are: 2*3*5*6, 2*3*30, 2*6*15, 3*6*10, 6*30. - _Gus Wiseman_, Oct 10 2017
		

Links

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

Crossrefs

Cf. A001055, A005117, A045778, A046523, A050320, A050327, a(p^k)=0 (p>1), a(A002110)=A000110, a(n!)=A103775(n), A206778, A293243.

Programs

Haskell

import Data.List (subsequences, genericIndex)
a050326 n = genericIndex a050326_list (n-1)
a050326_list = 1 : f 2 where
   f x = (if x /= s then a050326 s
                    else length $ filter (== x) $ map product $
                         subsequences $ tail $ a206778_row x) : f (x + 1)
         where s = a046523 x
-- Reinhard Zumkeller, May 03 2013

Maple

N:= 1000: # to get a(1)..a(N)
A:= Vector(N):
A[1]:= 1:
for n from 2 to N do
  if numtheory:-issqrfree(n) then
     S:= [$1..N/n]; T:= n*S; A[T]:= A[T]+A[S]
    fi;
od:
convert(A,list); # Robert Israel, Oct 10 2017

Mathematica

sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[sqfacs[n]],{n,100}] (* Gus Wiseman, Oct 10 2017 *)

Formula

Dirichlet g.f.: prod{n is squarefree and > 1}(1+1/n^s).

a(n) = A050327(A101296(n)). - R. J. Mathar, May 26 2017