cp's OEIS Frontend

A348381 Number of inseparable factorizations of n that are not a twin (x*x).

%I A348381 #7 Nov 03 2021 09:08:04
%S A348381 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 A348381 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 A348381 0,0,0,1,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0
%N A348381 Number of inseparable factorizations of n that are not a twin (x*x).
%C A348381 First differs from A347706 at a(216) = 3, A347706(216) = 4.
%C A348381 A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
%C A348381 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 A348381 a(n > 1) = A333487(n) - A010052(n).
%F A348381 a(2^n) = A325535(n) - 1 for odd n, otherwise A325535(n).
%e A348381 The a(n) factorizations for n = 96, 192, 384, 576:
%e A348381   2*2*2*12      3*4*4*4         4*4*4*6           4*4*4*9
%e A348381   2*2*2*2*6     2*2*2*24        2*2*2*48          2*2*2*72
%e A348381   2*2*2*2*2*3   2*2*2*2*12      2*2*2*2*24        2*2*2*2*36
%e A348381                 2*2*2*2*2*6     2*2*2*2*3*8       2*2*2*2*4*9
%e A348381                 2*2*2*2*3*4     2*2*2*2*4*6       2*2*2*2*6*6
%e A348381                 2*2*2*2*2*2*3   2*2*2*2*2*12      2*2*2*2*2*18
%e A348381                                 2*2*2*2*2*2*6     2*2*2*2*3*12
%e A348381                                 2*2*2*2*2*3*4     2*2*2*2*2*2*9
%e A348381                                 2*2*2*2*2*2*2*3   2*2*2*2*2*3*6
%e A348381                                                   2*2*2*2*2*2*3*3
%t A348381 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A348381 Table[Length[Select[facs[n],!MatchQ[#,{x_,x_}]&&Select[Permutations[#],!MatchQ[#,{___,x_,x_,___}]&]=={}&]],{n,100}]
%Y A348381 Positions of nonzero terms are A046099.
%Y A348381 Partitions not of this type are counted by A325534 - A000035.
%Y A348381 Partitions of this type are counted by A325535 - A000035.
%Y A348381 Allowing twins gives A333487.
%Y A348381 The case without an alternating permutation is A347706, with twins A348380.
%Y A348381 The complement is counted by A348383, without twins A335434.
%Y A348381 A001055 counts factorizations, strict A045778, ordered A074206.
%Y A348381 A001250 counts alternating permutations of sets.
%Y A348381 A008480 counts permutations of prime indices, strict A335489.
%Y A348381 A025047 counts alternating or wiggly compositions.
%Y A348381 A335452 counts anti-run permutations of prime indices, complement A336107.
%Y A348381 A339846 counts even-length factorizations.
%Y A348381 A339890 counts odd-length factorizations.
%Y A348381 A344654 counts non-twin partitions without an alternating permutation.
%Y A348381 A348382 counts non-anti-run compositions that are not a twin.
%Y A348381 A348611 counts anti-run ordered factorizations.
%Y A348381 Cf. A038548, A336102, A344614, A344653, A344740, A347050, A347437, A347438, A347456, A348379, A348609.
%K A348381 nonn
%O A348381 1,32
%A A348381 _Gus Wiseman_, Oct 30 2021