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%I A326334 #4 Jun 28 2019 21:14:33
%S A326334 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 A326334 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 A326334 4,2,4,1,14,1,2,4,4,2,4,1,12,5,2,1,8,2,2
%N A326334 Number of sortable factorizations of n.
%C A326334 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 A326334 A001055(n) = a(n) + A326291(n).
%e A326334 The a(180) = 16 sortable factorizations:
%e A326334   (2*2*3*3*5)  (2*2*5*9)   (4*5*9)   (2*90)   (180)
%e A326334                (2*3*5*6)   (2*2*45)  (4*45)
%e A326334                (3*3*4*5)   (2*5*18)  (5*36)
%e A326334                (2*2*3*15)  (2*6*15)  (12*15)
%e A326334                            (3*4*15)
%e A326334                            (3*5*12)
%e A326334 Missing from this list are the following unsortable factorizations:
%e A326334   (2*3*3*10)  (5*6*6)   (3*60)
%e A326334               (2*3*30)  (6*30)
%e A326334               (2*9*10)  (9*20)
%e A326334               (3*3*20)  (10*18)
%e A326334               (3*6*10)
%t A326334 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A326334 Table[Length[Select[facs[n],OrderedQ[Join@@Sort[First/@FactorInteger[#]&/@#,OrderedQ[PadRight[{#1,#2}]]&]]&]],{n,100}]
%Y A326334 Factorizations are A001055.
%Y A326334 Unsortable factorizations are A326291.
%Y A326334 Sortable integer partitions are A326333.
%Y A326334 Cf. A058681, A326211, A326212, A326237, A326258, A326332.
%K A326334 nonn
%O A326334 1,4
%A A326334 _Gus Wiseman_, Jun 27 2019

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