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A050326
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Number of factorizations of n into distinct squarefree numbers > 1.
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40
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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
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OFFSET
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1,6
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COMMENTS
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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).
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
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LINKS
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FORMULA
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Dirichlet g.f.: prod{n is squarefree and > 1}(1+1/n^s).
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EXAMPLE
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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
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A001055, A005117, A045778, A046523, A050320, A050327, a(p^k)=0 (p>1), a(A002110)=A000110, a(n!)=A103775(n), A206778, A293243.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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