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proposed

allocated for Gus WisemanNumber 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

1,4

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.

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

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)

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}]

Factorizations are A001055.

Unsortable factorizations are A326291.

Sortable integer partitions are A326333.

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

allocated

nonn

Gus Wiseman, Jun 27 2019

approved

editing

allocated for Gus Wiseman

allocated

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