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%I #8 Oct 27 2021 22:24:26
%S 0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,1,0,1,0,0,0,2,0,0,1,1,0,1,0,2,0,0,
%T 0,2,0,0,0,2,0,1,0,1,1,0,0,3,0,1,0,1,0,1,0,2,0,0,0,4,0,0,1,2,0,1,0,1,
%U 0,1,0,4,0,0,1,1,0,1,0,3,1,0,0,4,0,0,0
%N Number of distinct rational numbers of the form x * z / y for some factorization x * y * z = n, 1 < x <= y <= z.
%C This is also the number of distinct possible alternating products of length-3 factorizations of n, where we define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)), and where a factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
%e Representative factorizations for each of the a(360) = 9 alternating products:
%e (2,2,90) -> 90
%e (2,3,60) -> 40
%e (2,4,45) -> 45/2
%e (2,5,36) -> 72/5
%e (2,6,30) -> 10
%e (2,9,20) -> 40/9
%e (2,10,18) -> 18/5
%e (2,12,15) -> 5/2
%e (3,8,15) -> 45/8
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t Table[Length[Union[altprod/@Select[facs[n],Length[#]==3&]]],{n,100}]
%Y Allowing factorizations of any length <= 3 gives A033273.
%Y Positions of positive terms are A033942.
%Y Positions of 0's are A037143.
%Y The length-2 version is A072670.
%Y Allowing any length gives A347460, reverse A038548.
%Y Allowing any odd length gives A347708.
%Y A001055 counts factorizations (strict A045778, ordered A074206).
%Y A122179 counts length-3 factorizations.
%Y A292886 counts knapsack factorizations, by sum A293627.
%Y A301957 counts distinct subset-products of prime indices.
%Y A304792 counts distinct subset-sums of partitions, positive A276024.
%Y Cf. A000040, A001358, A002033, A046951, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456, A347461.
%K nonn
%O 1,24
%A _Gus Wiseman_, Oct 14 2021
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