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A349059
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Number of weakly alternating ordered factorizations of n.
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17
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1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, 1, 3, 3, 8, 1, 8, 1, 8, 3, 3, 1, 18, 2, 3, 4, 8, 1, 11, 1, 16, 3, 3, 3, 22, 1, 3, 3, 18, 1, 11, 1, 8, 8, 3, 1, 38, 2, 8, 3, 8, 1, 18, 3, 18, 3, 3, 1, 32, 1, 3, 8, 28, 3, 11, 1, 8, 3, 11, 1, 56, 1, 3, 8, 8, 3, 11, 1, 38, 8, 3
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OFFSET
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1,4
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COMMENTS
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An ordered factorization of n is a finite sequence of positive integers > 1 with product n.
We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.
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LINKS
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FORMULA
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EXAMPLE
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The ordered factorizations for n = 2, 4, 6, 8, 12, 24, 30:
(2) (4) (6) (8) (12) (24) (30)
(2*2) (2*3) (2*4) (2*6) (3*8) (5*6)
(3*2) (4*2) (3*4) (4*6) (6*5)
(2*2*2) (4*3) (6*4) (10*3)
(6*2) (8*3) (15*2)
(2*2*3) (12*2) (2*15)
(2*3*2) (2*12) (3*10)
(3*2*2) (2*2*6) (2*5*3)
(2*4*3) (3*2*5)
(2*6*2) (3*5*2)
(3*2*4) (5*2*3)
(3*4*2)
(4*2*3)
(6*2*2)
(2*2*2*3)
(2*2*3*2)
(2*3*2*2)
(3*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]]}]];
whkQ[y_]:=And@@Table[If[EvenQ[m], y[[m]]<=y[[m+1]], y[[m]]>=y[[m+1]]], {m, 1, Length[y]-1}];
Table[Length[Select[Join@@Permutations/@facs[n], whkQ[#]||whkQ[-#]&]], {n, 100}]
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CROSSREFS
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As compositions these are ranked by the complement of A349057.
A345164 counts alternating permutations of prime factors, w/ twins A344606.
A345170 counts partitions with an alternating permutation.
A348379 = factorizations w/ alternating permutation, complement A348380.
A349800 = weakly but not strongly alternating compositions, ranked A349799.
Cf. A003242, A122181, A138364, A339846, A339890, A345165, A345167, A345194, A347050, A347438, A347463, A347706.
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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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