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%I #4 Jun 28 2019 21:14:33
%S 1,1,1,2,1,2,1,3,2,2,1,4,1,2,2,5,1,4,1,4,2,2,1,7,2,2,3,4,1,4,1,7,2,2,
%T 2,8,1,2,2,7,1,4,1,4,4,2,1,12,2,4,2,4,1,7,2,7,2,2,1,8,1,2,4,11,2,4,1,
%U 4,2,4,1,14,1,2,4,4,2,4,1,12,5,2,1,8,2,2
%N Number of sortable factorizations of n.
%C A factorization into factors > 1 is sortable if there is a permutation (c_1,...,c_k) of the factors such that the maximum prime factor (in the standard factorization of an integer into prime numbers) of c_i is at most the minimum prime factor of c_{i+1}. For example, the factorization (6*8*27) is sortable because the permutation (8,6,27) satisfies the condition.
%F A001055(n) = a(n) + A326291(n).
%e The a(180) = 16 sortable factorizations:
%e (2*2*3*3*5) (2*2*5*9) (4*5*9) (2*90) (180)
%e (2*3*5*6) (2*2*45) (4*45)
%e (3*3*4*5) (2*5*18) (5*36)
%e (2*2*3*15) (2*6*15) (12*15)
%e (3*4*15)
%e (3*5*12)
%e Missing from this list are the following unsortable factorizations:
%e (2*3*3*10) (5*6*6) (3*60)
%e (2*3*30) (6*30)
%e (2*9*10) (9*20)
%e (3*3*20) (10*18)
%e (3*6*10)
%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],OrderedQ[Join@@Sort[First/@FactorInteger[#]&/@#,OrderedQ[PadRight[{#1,#2}]]&]]&]],{n,100}]
%Y Factorizations are A001055.
%Y Unsortable factorizations are A326291.
%Y Sortable integer partitions are A326333.
%Y Cf. A058681, A326211, A326212, A326237, A326258, A326332.
%K nonn
%O 1,4
%A _Gus Wiseman_, Jun 27 2019
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