A348383 - OEIS

%I #8 Nov 03 2021 09:08:16

%S 1,1,1,2,1,2,1,2,2,2,1,4,1,2,2,4,1,4,1,4,2,2,1,6,2,2,2,4,1,5,1,5,2,2,

%T 2,9,1,2,2,6,1,5,1,4,4,2,1,10,2,4,2,4,1,6,2,6,2,2,1,11,1,2,4,7,2,5,1,

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

%N Number of factorizations of n that are either separable (have an anti-run permutation) or are a twin (x*x).

%C First differs from A347050 at a(216) = 28, A347050(216) = 27.

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

%C A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts. Alternatively, a multiset is separable if its greatest multiplicity is greater than the sum of the remaining multiplicities plus one.

%F a(n > 1) = A335434(n) + A010052(n), where A010052(n) = 1 if n is a perfect square, otherwise 0.

%e The a(216) = 28 factorizations:

%e   (2*2*2*3*3*3)  (2*2*2*3*9)  (2*2*6*9)   (3*8*9)   (3*72)   (216)

%e                  (2*2*3*3*6)  (2*3*4*9)   (4*6*9)   (4*54)

%e                  (2*3*3*3*4)  (2*3*6*6)   (2*2*54)  (6*36)

%e                               (3*3*4*6)   (2*3*36)  (8*27)

%e                               (2*2*3*18)  (2*4*27)  (9*24)

%e                               (2*3*3*12)  (2*6*18)  (12*18)

%e                                           (2*9*12)  (2*108)

%e                                           (3*3*24)

%e                                           (3*4*18)

%e                                           (3*6*12)

%e The a(270) = 20 factorizations:

%e   (2*3*3*3*5)  (2*3*5*9)   (5*6*9)   (3*90)   (270)

%e                (3*3*5*6)   (2*3*45)  (5*54)

%e                (2*3*3*15)  (2*5*27)  (6*45)

%e                            (2*9*15)  (9*30)

%e                            (3*3*30)  (10*27)

%e                            (3*5*18)  (15*18)

%e                            (3*6*15)  (2*135)

%e                            (3*9*10)

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

%t sepQ[m_]:=Select[Permutations[m],!MatchQ[#,{___,x_,x_,___}]&]!={};

%t Table[Length[Select[facs[n],MatchQ[#,{x_,x_}]||sepQ[#]&]],{n,100}]

%Y Positions of 1's are 1 and A000040.

%Y Not requiring separability gives A010052 for n > 1.

%Y Positions of 2's are A323644.

%Y Partitions of this type are counted by A325534(n) + A000035(n + 1).

%Y Partitions of this type are ranked by A335433 \/ A001248.

%Y Partitions not of this type are counted by A325535(n) - A000035(n + 1).

%Y Partitions not of this type are ranked by A345193 = A335448 \ A001248.

%Y Not allowing twins gives A335434, complement A333487,

%Y The case with an alternating permutation is A347050, no twins A348379.

%Y The case without an alternating permutation is A347706, no twins A348380.

%Y The complement is counted by A348381.

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

%Y A001250 counts alternating permutations.

%Y A003242 counts anti-run compositions, ranked by A333489.

%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 Cf. A119620, A138364, A336103, A344654, A344740, A347437, A347438, A347456, A347458, A348382.

%K nonn

%O 1,4

%A _Gus Wiseman_, Oct 30 2021