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%I #8 May 02 2018 09:24:40
%S 0,1,1,0,1,2,1,0,0,2,1,3,1,2,2,0,1,3,1,3,2,2,1,4,0,2,0,3,1,5,1,0,2,2,
%T 2,3,1,2,2,4,1,5,1,3,3,2,1,5,0,3,2,3,1,4,2,4,2,2,1,9,1,2,3,0,2,5,1,3,
%U 2,5,1,8,1,2,3,3,2,5,1,5,0,2,1,9,2,2,2,4,1,9,2
%N Number of aperiodic factorizations of n using elements of A007916 (numbers that are not perfect powers).
%C An aperiodic factorization of n is a finite multiset of positive integers greater than 1 whose product is n and whose multiplicities are relatively prime.
%C The positions of zeros in this sequence are the prime powers A000961.
%F a(n) = Sum_{d in A007916, d|A052409(n)} mu(d) * A303707(n^(1/d)).
%e The a(144) = 8 aperiodic factorizations are (2*2*2*3*6), (2*2*2*18), (2*2*3*12), (2*3*24), (2*6*12), (2*72), (3*48) and (6*24). Missing from this list are (12*12), (2*2*6*6) and (2*2*2*2*3*3).
%t radQ[n_]:=Or[n===1,GCD@@FactorInteger[n][[All,2]]===1];
%t facsr[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsr[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],radQ]}]];
%t Table[Length[Select[facsr[n],GCD@@Length/@Split[#]===1&]],{n,100}]
%Y Cf. A000740, A000837, A001055, A001597, A007716, A007916, A052409, A052410, A100953, A275870, A281116, A301700, A303386, A303431, A303546, A303551, A303707, A303709, A303710.
%K nonn
%O 1,6
%A _Gus Wiseman_, Apr 29 2018
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