A335434 - OEIS

A335434

Number of separable factorizations of n into factors > 1.

%I #6 Jul 04 2020 09:22:49

%S 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,

%T 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,

%U 4,2,5,1,15,1,2,4,4,2,5,1,10,3,2,1,11,2

%N Number of separable 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 A333487(n) + a(n) = A001055(n).

%e The a(n) factorizations for n = 2, 6, 16, 12, 30, 24, 36, 48, 60:

%e 2 6 16 12 30 24 36 48 60

%e 2*3 2*8 2*6 5*6 3*8 4*9 6*8 2*30

%e 2*2*4 3*4 2*15 4*6 2*18 2*24 3*20

%e 2*2*3 3*10 2*12 3*12 3*16 4*15

%e 2*3*5 2*2*6 2*2*9 4*12 5*12

%e 2*3*4 2*3*6 2*3*8 6*10

%e 3*3*4 2*4*6 2*5*6

%e 2*2*3*3 3*4*4 3*4*5

%e 2*2*12 2*2*15

%e 2*2*3*4 2*3*10

%e 2*2*3*5

%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 A325534.

%Y The inseparable version is A333487.

%Y The version for multisets with prescribed multiplicities is A335127.

%Y Factorizations are A001055.

%Y Anti-run compositions are A003242.

%Y Inseparable partitions are A325535.

%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 Cf. A106351, A292884, A295370, A333628, A333755, A335463, A335125, A335126, A335407, A335457, A335474, A335516, A335838.

%K nonn

%O 1,6

%A _Gus Wiseman_, Jul 03 2020