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A347456
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Number of factorizations of n with alternating product >= 1.
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27
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1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 2, 2, 1, 2, 1, 4, 1, 1, 1, 6, 1, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 6, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 2, 8, 1, 2, 1, 2, 1, 2, 1, 8, 1, 1, 2, 2, 1, 2, 1, 6, 4, 1, 1, 5, 1, 1, 1
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OFFSET
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1,4
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COMMENTS
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We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
Also the number of factorizations of n with alternating sum >= 0.
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LINKS
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FORMULA
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EXAMPLE
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The a(n) factorizations for n = 4, 16, 24, 36, 60, 64, 96:
4 16 24 36 60 64 96
2*2 4*4 2*2*6 6*6 2*5*6 8*8 2*6*8
2*2*4 2*3*4 2*2*9 3*4*5 2*4*8 3*4*8
2*2*2*2 2*3*6 2*2*15 4*4*4 4*4*6
3*3*4 2*3*10 2*2*16 2*2*24
2*2*3*3 2*2*4*4 2*3*16
2*2*2*2*4 2*4*12
2*2*2*2*2*2 2*2*2*2*6
2*2*2*3*4
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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[Select[facs[n], altprod[#]>=1&]], {n, 100}]
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CROSSREFS
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Positions of 3's appear to be A065036.
Positions of 1's are 1 and A167171.
The opposite version (<= instead of >=) is A339846.
The strict version (> instead of >=) is A339890, also the odd-length case.
Allowing any integer alternating product gives A347437.
The case of alternating product 1 is A347438, also the even-length case.
Allowing any integer reciprocal alternating product gives A347439.
The complement (< instead of >=) is A347440.
Allowing any integer reverse-alternating product gives A347442.
A038548 counts factorizations with a wiggly permutation.
A045778 counts strict factorizations.
A074206 counts ordered factorizations.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1.
A347447 counts strict factorizations with alternating product > 1.
Cf. A001700, A028983, A316523, A347441, A347443, A347446, A347448, A347450, A347454, A347463, A347708.
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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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