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%I #28 May 11 2019 02:13:01
%S 1,1,1,1,1,3,1,1,1,3,1,5,1,3,3,1,1,5,1,5,3,3,1,7,1,3,1,5,1,13,1,1,3,3,
%T 3,13,1,3,3,7,1,13,1,5,5,3,1,9,1,5,3,5,1,7,3,7,3,3,1,31,1,3,5,1,3,13,
%U 1,5,3,13,1,25,1,3,5,5,3,13,1,9,1,3,1,31,3,3,3,7,1,31,3,5,3,3,3,11,1,5
%N Number of ordered factorizations of n into 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).
%H Reinhard Zumkeller, <a href="/A050328/b050328.txt">Table of n, a(n) for n = 1..10000</a>
%F Dirichlet g.f.: 1/(1-B(s)) where B(s) is D.g.f. of characteristic function of squarefree numbers > 1.
%F a(A000961(n)) = 1.
%F a(A002110(n)) = A000670(n).
%F a(n) = Sum_{d divides n, d<n} A008966(n/d)*a(d). - _Vladeta Jovovic_, Sep 25 2002, corrected by _Antti Karttunen_, May 27 2017
%F G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} mu(k)^2*A(x^k). - _Ilya Gutkovskiy_, May 10 2019
%t a[n_]:=If[n==1, n, Sum[If[(d<n && SquareFreeQ[n/d]), a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 100}] (* _Indranil Ghosh_, May 27 2017 *)
%o (Haskell)
%o import Data.List (genericIndex)
%o a050328 n = genericIndex a050328_list (n-1)
%o a050328_list = f 1 where
%o f x = (if x == 1 then 1 else
%o sum $ map (a050328 . (div x)) $ tail $ a206778_row x) : f (x + 1)
%o -- _Reinhard Zumkeller_, May 03 2013
%o (PARI) A050328(n) = if(1==n,n,sumdiv(n,d,if((d<n && issquarefree(n/d)),A050328(d),0))); \\ _Antti Karttunen_, May 27 2017
%Y Cf. A000670, A000961, A002033, A002110, A005117, A008966, A050329, A114006, A206778.
%K nonn
%O 1,6
%A _Christian G. Bower_, Oct 15 1999
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