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A333487
Number of inseparable factorizations of n into factors > 1.
14
0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0
OFFSET
1,16
COMMENTS
A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.
FORMULA
a(n) + A335434(n) = A001055(n).
EXAMPLE
The a(n) factorizations for n = 4, 16, 96, 144, 64, 192:
2*2 4*4 2*2*2*12 12*12 8*8 3*4*4*4
2*2*2*2 2*2*2*2*6 2*2*2*18 4*4*4 2*2*2*24
2*2*2*2*2*3 2*2*2*2*9 2*2*2*8 2*2*2*2*12
2*2*2*2*3*3 2*2*2*2*4 2*2*2*2*2*6
2*2*2*2*2*2 2*2*2*2*3*4
2*2*2*2*2*2*3
MATHEMATICA
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}]
CROSSREFS
The version for partitions is A325535.
The version for multisets with prescribed multiplicities is A335126.
The separable version is A335434.
Anti-run compositions are A003242.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Patterns contiguously matched by compositions are A335457.
Sequence in context: A304819 A304820 A162641 * A348380 A185374 A332014
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 01 2020
STATUS
approved