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%I #8 Oct 27 2021 22:24:17
%S 0,1,1,1,1,1,1,2,1,1,1,2,1,1,1,2,1,2,1,2,1,1,1,3,1,1,2,2,1,2,1,3,1,1,
%T 1,3,1,1,1,3,1,2,1,2,2,1,1,4,1,2,1,2,1,2,1,3,1,1,1,5,1,1,2,3,1,2,1,2,
%U 1,2,1,5,1,1,2,2,1,2,1,4,2,1,1,5,1,1,1
%N Number of distinct possible alternating products of odd-length factorizations of n.
%C We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%C A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
%C Note that it is sufficient to look at only length-1 and length-3 factorizations; cf. A347709.
%F Conjecture: For n > 1, a(n) = 1 + A347460(n) - A038548(n) + A072670(n).
%e Representative factorizations for each of the a(180) = 7 alternating products:
%e (2*2*3*3*5) -> 5
%e (2*2*45) -> 45
%e (2*3*30) -> 20
%e (2*5*18) -> 36/5
%e (2*9*10) -> 20/9
%e (3*4*15) -> 45/4
%e (180) -> 180
%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],OddQ[Length[#]]&]]],{n,100}]
%Y The version for partitions is A028310, reverse A347707.
%Y Positions of 1's appear to be A037143 \ {1}.
%Y The even-length version for n > 1 is A072670, strict A211159.
%Y Counting only integers appears to give A293234, with evens A046951.
%Y This is the odd-length case of A347460, reverse A038548.
%Y The any-length version for partitions is A347461, reverse A347462.
%Y The length-3 case is A347709.
%Y A001055 counts factorizations (strict A045778, ordered A074206).
%Y A056239 adds up prime indices, row sums of A112798.
%Y A276024 counts distinct positive subset-sums of partitions.
%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.
%Y A347050 = factorizations w/ an alternating permutation, complement A347706.
%Y A347441 counts odd-length factorizations with integer alternating product.
%Y Cf. A002033, A103919, A108917, A119620, A325770, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456.
%K nonn
%O 1,8
%A _Gus Wiseman_, Oct 11 2021
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