A326334 Number of sortable factorizations of n.

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, 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, 4, 2, 4, 1, 14, 1, 2, 4, 4, 2, 4, 1, 12, 5, 2, 1, 8, 2, 2

Offset: 1

Author

Gus Wiseman, Jun 27 2019

Keywords

• nonn

Comments

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.

Examples

			The a(180) = 16 sortable factorizations:
  (2*2*3*3*5)  (2*2*5*9)   (4*5*9)   (2*90)   (180)
               (2*3*5*6)   (2*2*45)  (4*45)
               (3*3*4*5)   (2*5*18)  (5*36)
               (2*2*3*15)  (2*6*15)  (12*15)
                           (3*4*15)
                           (3*5*12)
Missing from this list are the following unsortable factorizations:
  (2*3*3*10)  (5*6*6)   (3*60)
              (2*3*30)  (6*30)
              (2*9*10)  (9*20)
              (3*3*20)  (10*18)
              (3*6*10)
		

Crossrefs

Factorizations are A001055.

Unsortable factorizations are A326291.

Sortable integer partitions are A326333.

Cf. A058681, A326211, A326212, A326237, A326258, A326332.

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],OrderedQ[Join@@Sort[First/@FactorInteger[#]&/@#,OrderedQ[PadRight[{#1,#2}]]&]]&]],{n,100}]

Formula

A001055(n) = a(n) + A326291(n).