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A348383
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Number of factorizations of n that are either separable (have an anti-run permutation) or are a twin (x*x).
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8
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1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 9, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 15, 1, 2, 4, 4, 2, 5, 1, 10, 4, 2, 1, 11, 2
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OFFSET
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1,4
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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.
A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts. Alternatively, a multiset is separable if its greatest multiplicity is greater than the sum of the remaining multiplicities plus one.
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LINKS
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FORMULA
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EXAMPLE
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The a(216) = 28 factorizations:
(2*2*2*3*3*3) (2*2*2*3*9) (2*2*6*9) (3*8*9) (3*72) (216)
(2*2*3*3*6) (2*3*4*9) (4*6*9) (4*54)
(2*3*3*3*4) (2*3*6*6) (2*2*54) (6*36)
(3*3*4*6) (2*3*36) (8*27)
(2*2*3*18) (2*4*27) (9*24)
(2*3*3*12) (2*6*18) (12*18)
(2*9*12) (2*108)
(3*3*24)
(3*4*18)
(3*6*12)
The a(270) = 20 factorizations:
(2*3*3*3*5) (2*3*5*9) (5*6*9) (3*90) (270)
(3*3*5*6) (2*3*45) (5*54)
(2*3*3*15) (2*5*27) (6*45)
(2*9*15) (9*30)
(3*3*30) (10*27)
(3*5*18) (15*18)
(3*6*15) (2*135)
(3*9*10)
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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]]}]];
sepQ[m_]:=Select[Permutations[m], !MatchQ[#, {___, x_, x_, ___}]&]!={};
Table[Length[Select[facs[n], MatchQ[#, {x_, x_}]||sepQ[#]&]], {n, 100}]
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CROSSREFS
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Positions of 1's are 1 and A000040.
Not requiring separability gives A010052 for n > 1.
Partitions not of this type are counted by A325535(n) - A000035(n + 1).
The case with an alternating permutation is A347050, no twins A348379.
The case without an alternating permutation is A347706, no twins A348380.
The complement is counted by A348381.
A001250 counts alternating permutations.
A025047 counts alternating or wiggly compositions.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
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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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