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A335434
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Number of separable factorizations of n into factors > 1.
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18
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1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 6, 1, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 1, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 6, 2, 5, 1, 4, 2, 5, 1, 15, 1, 2, 4, 4, 2, 5, 1, 10, 3, 2, 1, 11, 2
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OFFSET
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1,6
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COMMENTS
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A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.
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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, 6, 16, 12, 30, 24, 36, 48, 60:
2 6 16 12 30 24 36 48 60
2*3 2*8 2*6 5*6 3*8 4*9 6*8 2*30
2*2*4 3*4 2*15 4*6 2*18 2*24 3*20
2*2*3 3*10 2*12 3*12 3*16 4*15
2*3*5 2*2*6 2*2*9 4*12 5*12
2*3*4 2*3*6 2*3*8 6*10
3*3*4 2*4*6 2*5*6
2*2*3*3 3*4*4 3*4*5
2*2*12 2*2*15
2*2*3*4 2*3*10
2*2*3*5
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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]]}]];
Table[Length[Select[facs[n], Select[Permutations[#], !MatchQ[#, {___, x_, x_, ___}]&]!={}&]], {n, 100}]
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CROSSREFS
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The version for partitions is A325534.
The inseparable version is A333487.
The version for multisets with prescribed multiplicities is A335127.
Inseparable partitions are A325535.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Cf. A106351, A292884, A295370, A333628, A333755, A335463, A335125, A335126, A335407, A335457, A335474, A335516, A335838.
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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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