A050326 - OEIS

%I #24 Oct 10 2017 04:09:02

%S 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,

%T 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,

%U 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

%N Number of factorizations of n into distinct squarefree numbers > 1.

%C 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).

%C 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

%C 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

%H Reinhard Zumkeller, <a href="/A050326/b050326.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(n) = A050327(A101296(n)). - _R. J. Mathar_, May 26 2017

%e 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

%p N:= 1000: # to get a(1)..a(N)

%p A:= Vector(N):

%p A[1]:= 1:

%p for n from 2 to N do

%p   if numtheory:-issqrfree(n) then

%p      S:= [$1..N/n]; T:= n*S; A[T]:= A[T]+A[S]

%p     fi;

%p od:

%p convert(A,list); # _Robert Israel_, Oct 10 2017

%t sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];

%t Table[Length[sqfacs[n]],{n,100}] (* _Gus Wiseman_, Oct 10 2017 *)

%o (Haskell)

%o import Data.List (subsequences, genericIndex)

%o a050326 n = genericIndex a050326_list (n-1)

%o a050326_list = 1 : f 2 where

%o    f x = (if x /= s then a050326 s

%o                     else length $ filter (== x) $ map product $

%o                          subsequences $ tail $ a206778_row x) : f (x + 1)

%o          where s = a046523 x

%o -- _Reinhard Zumkeller_, May 03 2013

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

%K nonn

%O 1,6

%A _Christian G. Bower_, Oct 15 1999