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A347459
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Number of factorizations of n^2 with integer reciprocal alternating product.
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10
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1, 1, 1, 3, 1, 4, 1, 6, 3, 4, 1, 11, 1, 4, 4, 12, 1, 11, 1, 12, 4, 4, 1, 28, 3, 4, 6, 12, 1, 19, 1, 22, 4, 4, 4, 38, 1, 4, 4, 29, 1, 21, 1, 12, 11, 4, 1, 65, 3, 11, 4, 12, 1, 29, 4, 29, 4, 4, 1, 71, 1, 4, 11, 40, 4, 22, 1, 12, 4, 18, 1, 107, 1, 4, 11, 12, 4
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OFFSET
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1,4
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COMMENTS
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We define the reciprocal alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^i).
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
All such factorizations have even length.
Image appears to be 1, 3, 4, 6, 11, ... , missing some numbers such as 2, 5, 7, 8, 9, ...
The case of alternating product 1, the case of alternating sum 0, and the reverse version are all counted by A001055.
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LINKS
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FORMULA
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EXAMPLE
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The a(2) = 1 through a(10) = 4 factorizations:
2*2 3*3 2*8 5*5 6*6 7*7 8*8 9*9 2*50
4*4 2*18 2*32 3*27 5*20
2*2*2*2 3*12 4*16 3*3*3*3 10*10
2*2*3*3 2*2*2*8 2*2*5*5
2*2*4*4
2*2*2*2*2*2
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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]]}]];
recaltprod[q_]:=Product[q[[i]]^(-1)^i, {i, Length[q]}];
Table[Length[Select[facs[n^2], IntegerQ[recaltprod[#]]&]], {n, 100}]
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CROSSREFS
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The additive version (partitions) is A000041, the even bisection of A119620.
The restriction to powers of 2 is A236913, the even bisection of A027187.
The nonsquared nonreciprocal even-length version is A347438.
This is the restriction to perfect squares of A347439.
A046099 counts factorizations with no alternating permutations.
A273013 counts ordered factorizations of n^2 with alternating product 1.
A347460 counts possible alternating products of factorizations.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A347457 ranks partitions with integer alternating product.
A347466 counts factorizations of n^2.
Cf. A062312, A316523, A330972, A344653, A346635, A347440, A347441, A347442, A347453, A347461, A347463, A347464, A347705.
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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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