A333487 - OEIS

A333487

Number of inseparable factorizations of n into factors > 1.

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

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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.

LINKS

Table of n, a(n) for n=1..87.

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.

Cf. A106351, A292884, A295370, A333628, A333755, A335463, A335125, A335127, A335407, A335474, A335516, A335838.

Sequence in context: A304819 A304820 A162641 * A348380 A185374 A332014

Adjacent sequences: A333484 A333485 A333486 * A333488 A333489 A333490

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 01 2020

STATUS

approved