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A347709
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Number of distinct rational numbers of the form x * z / y for some factorization x * y * z = n, 1 < x <= y <= z.
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2
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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, 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, 0, 1, 0, 4, 0, 0, 1, 1, 0, 1, 0, 3, 1, 0, 0, 4, 0, 0, 0
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OFFSET
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1,24
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COMMENTS
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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.
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LINKS
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EXAMPLE
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Representative factorizations for each of the a(360) = 9 alternating products:
(2,2,90) -> 90
(2,3,60) -> 40
(2,4,45) -> 45/2
(2,5,36) -> 72/5
(2,6,30) -> 10
(2,9,20) -> 40/9
(2,10,18) -> 18/5
(2,12,15) -> 5/2
(3,8,15) -> 45/8
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MATHEMATICA
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facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1), {i, Length[q]}];
Table[Length[Union[altprod/@Select[facs[n], Length[#]==3&]]], {n, 100}]
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CROSSREFS
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Allowing factorizations of any length <= 3 gives A033273.
Positions of positive terms are A033942.
Allowing any odd length gives A347708.
A122179 counts length-3 factorizations.
A301957 counts distinct subset-products of prime indices.
Cf. A000040, A001358, A002033, A046951, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456, A347461.
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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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