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A347441
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Number of odd-length factorizations of n with integer alternating product.
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17
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0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 5, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 2, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 6, 1, 2, 2, 4, 1, 1, 1, 2, 1, 1, 1, 7
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OFFSET
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1,8
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COMMENTS
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A factorization of n is a weakly increasing 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) factorizations for n = 2, 8, 32, 48, 54, 72, 108:
2 8 32 48 54 72 108
2*2*2 2*2*8 2*4*6 2*3*9 2*6*6 2*6*9
2*4*4 3*4*4 3*3*6 3*3*8 3*6*6
2*2*2*2*2 2*2*12 2*2*18 2*2*27
2*2*2*2*3 2*3*12 2*3*18
2*2*2*3*3 3*3*12
2*2*3*3*3
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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], OddQ[Length[#]]&&IntegerQ[altprod[#]]&]], {n, 100}]
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PROG
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(PARI) A347441(n, m=n, ap=1, e=0) = if(1==n, (e%2)&&1==denominator(ap), sumdiv(n, d, if((d>1)&&(d<=m), A347441(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Oct 22 2023
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CROSSREFS
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The restriction to powers of 2 is A027193.
Allowing any alternating product gives A339890.
Allowing even-length factorizations gives A347437.
The even-length instead of odd-length version is A347438.
A038548 counts possible reverse-alternating products of factorizations.
A273013 counts ordered factorizations of n^2 with alternating product 1.
A339846 counts even-length factorizations.
A347439 counts factorizations with integer reciprocal alternating product.
A347440 counts factorizations with alternating product < 1.
A347442 counts factorizations with integer reverse-alternating product.
A347456 counts factorizations with alternating product >= 1.
A347463 counts ordered factorizations with integer alternating product.
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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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