A348381 - OEIS

%I #7 Nov 03 2021 09:08:04

%S 0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,2,0,0,

%T 0,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,4,0,0,0,0,

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

%N Number of inseparable factorizations of n that are not a twin (x*x).

%C First differs from A347706 at a(216) = 3, A347706(216) = 4.

%C A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

%C A multiset is inseparable if it has no permutation that is an anti-run, meaning there are always adjacent equal parts. Alternatively, a multiset is inseparable if its maximal multiplicity is at most one plus the sum of its remaining multiplicities.

%F a(n > 1) = A333487(n) - A010052(n).

%F a(2^n) = A325535(n) - 1 for odd n, otherwise A325535(n).

%e The a(n) factorizations for n = 96, 192, 384, 576:

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

%e   2*2*2*2*6     2*2*2*24        2*2*2*48          2*2*2*72

%e   2*2*2*2*2*3   2*2*2*2*12      2*2*2*2*24        2*2*2*2*36

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

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

%e                 2*2*2*2*2*2*3   2*2*2*2*2*12      2*2*2*2*2*18

%e                                 2*2*2*2*2*2*6     2*2*2*2*3*12

%e                                 2*2*2*2*2*3*4     2*2*2*2*2*2*9

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

%e                                                   2*2*2*2*2*2*3*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],!MatchQ[#,{x_,x_}]&&Select[Permutations[#],!MatchQ[#,{___,x_,x_,___}]&]=={}&]],{n,100}]

%Y Positions of nonzero terms are A046099.

%Y Partitions not of this type are counted by A325534 - A000035.

%Y Partitions of this type are counted by A325535 - A000035.

%Y Allowing twins gives A333487.

%Y The case without an alternating permutation is A347706, with twins A348380.

%Y The complement is counted by A348383, without twins A335434.

%Y A001055 counts factorizations, strict A045778, ordered A074206.

%Y A001250 counts alternating permutations of sets.

%Y A008480 counts permutations of prime indices, strict A335489.

%Y A025047 counts alternating or wiggly compositions.

%Y A335452 counts anti-run permutations of prime indices, complement A336107.

%Y A339846 counts even-length factorizations.

%Y A339890 counts odd-length factorizations.

%Y A344654 counts non-twin partitions without an alternating permutation.

%Y A348382 counts non-anti-run compositions that are not a twin.

%Y A348611 counts anti-run ordered factorizations.

%Y Cf. A038548, A336102, A344614, A344653, A344740, A347050, A347437, A347438, A347456, A348379, A348609.

%K nonn

%O 1,32

%A _Gus Wiseman_, Oct 30 2021