%I #6 Jul 01 2020 22:30:16

%S 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,

%T 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,

%U 0,0,0,1,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0

%N Number of inseparable factorizations of n into factors > 1.

%C A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.

%F a(n) + A335434(n) = A001055(n).

%e The a(n) factorizations for n = 4, 16, 96, 144, 64, 192:

%e 2*2 4*4 2*2*2*12 12*12 8*8 3*4*4*4

%e 2*2*2*2 2*2*2*2*6 2*2*2*18 4*4*4 2*2*2*24

%e 2*2*2*2*2*3 2*2*2*2*9 2*2*2*8 2*2*2*2*12

%e 2*2*2*2*3*3 2*2*2*2*4 2*2*2*2*2*6

%e 2*2*2*2*2*2 2*2*2*2*3*4

%e 2*2*2*2*2*2*3

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Table[Length[Select[facs[n],Select[Permutations[#],!MatchQ[#,{___,x_,x_,___}]&]=={}&]],{n,100}]

%Y The version for partitions is A325535.

%Y The version for multisets with prescribed multiplicities is A335126.

%Y The separable version is A335434.

%Y Anti-run compositions are A003242.

%Y Anti-runs are ranked by A333489.

%Y Separable partitions are ranked by A335433.

%Y Inseparable partitions are ranked by A335448.

%Y Anti-run permutations of prime indices are A335452.

%Y Patterns contiguously matched by compositions are A335457.

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

%K nonn

%O 1,16

%A _Gus Wiseman_, Jul 01 2020