cp's OEIS Frontend

A335434 Number of separable factorizations of n into factors > 1.

%I A335434 #6 Jul 04 2020 09:22:49
%S A335434 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 A335434 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 A335434 4,2,5,1,15,1,2,4,4,2,5,1,10,3,2,1,11,2
%N A335434 Number of separable factorizations of n into factors > 1.
%C A335434 A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.
%F A335434 A333487(n) + a(n) = A001055(n).
%e A335434 The a(n) factorizations for n = 2, 6, 16, 12, 30, 24, 36, 48, 60:
%e A335434   2  6    16     12     30     24     36       48       60
%e A335434      2*3  2*8    2*6    5*6    3*8    4*9      6*8      2*30
%e A335434           2*2*4  3*4    2*15   4*6    2*18     2*24     3*20
%e A335434                  2*2*3  3*10   2*12   3*12     3*16     4*15
%e A335434                         2*3*5  2*2*6  2*2*9    4*12     5*12
%e A335434                                2*3*4  2*3*6    2*3*8    6*10
%e A335434                                       3*3*4    2*4*6    2*5*6
%e A335434                                       2*2*3*3  3*4*4    3*4*5
%e A335434                                                2*2*12   2*2*15
%e A335434                                                2*2*3*4  2*3*10
%e A335434                                                         2*2*3*5
%t A335434 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A335434 Table[Length[Select[facs[n],Select[Permutations[#],!MatchQ[#,{___,x_,x_,___}]&]!={}&]],{n,100}]
%Y A335434 The version for partitions is A325534.
%Y A335434 The inseparable version is A333487.
%Y A335434 The version for multisets with prescribed multiplicities is A335127.
%Y A335434 Factorizations are A001055.
%Y A335434 Anti-run compositions are A003242.
%Y A335434 Inseparable partitions are A325535.
%Y A335434 Anti-runs are ranked by A333489.
%Y A335434 Separable partitions are ranked by A335433.
%Y A335434 Inseparable partitions are ranked by A335448.
%Y A335434 Anti-run permutations of prime indices are A335452.
%Y A335434 Cf. A106351, A292884, A295370, A333628, A333755, A335463, A335125, A335126, A335407, A335457, A335474, A335516, A335838.
%K A335434 nonn
%O A335434 1,6
%A A335434 _Gus Wiseman_, Jul 03 2020