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A347049
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Number of odd-length ordered factorizations of n with integer alternating product.
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3
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0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 2, 3, 1, 1, 1, 7, 1, 1, 1, 11, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 14, 1, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 7, 1, 1, 3, 15, 1, 1, 1, 3, 1, 1, 1, 24, 1, 1, 3, 3, 1, 1, 1, 14, 4, 1, 1, 7, 1, 1, 1, 5, 1, 7, 1, 3, 1, 1, 1, 24, 1, 3, 3, 11
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OFFSET
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1,8
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COMMENTS
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An ordered factorization of n is a sequence of positive integers > 1 with product n.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
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LINKS
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FORMULA
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EXAMPLE
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The a(n) ordered factorizations for n = 2, 8, 12, 16, 24, 32, 36, 48:
2 8 12 16 24 32 36 48
2*2*2 2*2*3 2*2*4 2*2*6 2*2*8 2*2*9 2*4*6
3*2*2 2*4*2 3*2*4 2*4*4 2*3*6 3*2*8
4*2*2 4*2*3 4*2*4 2*6*3 3*4*4
6*2*2 4*4*2 3*2*6 4*2*6
8*2*2 3*3*4 4*4*3
2*2*2*2*2 3*6*2 6*2*4
4*3*3 6*4*2
6*2*3 8*2*3
6*3*2 12*2*2
9*2*2 2*2*12
2*2*2*2*3
2*2*3*2*2
3*2*2*2*2
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MATHEMATICA
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ordfacs[n_]:=If[n<=1, {{}}, Join@@Table[Prepend[#, d]&/@ordfacs[n/d], {d, Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1), {i, Length[q]}];
Table[Length[Select[ordfacs[n], OddQ[Length[#]]&&IntegerQ[altprod[#]]&]], {n, 100}]
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PROG
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(PARI) A347049(n, m=n, ap=1, e=0) = if(1==n, (e%2) && 1==denominator(ap), sumdiv(n, d, if(d>1, A347049(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Jul 28 2024
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CROSSREFS
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Positions of 2's appear to be A030078.
Positions of 3's appear to be A054753.
Positions of 1's appear to be A167207.
Allowing non-integer alternating product gives A174726, unordered A339890.
The even-length version is A347048.
The unordered version is A347441, with same reverse version.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1, ranked by A028982.
A347050 = factorizations with alternating permutation, complement A347706.
A347437 = factorizations with integer alternating product, reverse A347442.
A347438 = factorizations with alternating product 1, on squares A273013.
A347439 = factorizations with integer reciprocal alternating product.
A347446 = partitions with integer alternating product, reverse A347445.
A347457 lists Heinz numbers of partitions with integer alternating product.
A347460 counts possible alternating products of factorizations.
A347708 counts possible alternating products of odd-length factorizations.
Cf. A025047, A035363, A038548, A116406, A347440, A347454, A347456, A347458, A347459, A347464, A347704.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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