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A326334
Number of sortable factorizations of n.
2
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
1,4
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.
FORMULA
A001055(n) = a(n) + A326291(n).
EXAMPLE
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)
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}]
CROSSREFS
Factorizations are A001055.
Unsortable factorizations are A326291.
Sortable integer partitions are A326333.
Sequence in context: A317141 A317791 A318559 * A218320 A355030 A305254
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 27 2019
STATUS
approved