A326333 -id:A326333 - cp's OEIS frontend

A326332 Number of integer partitions of n with unsortable prime factors.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 5, 9, 14, 22, 33, 50, 71, 100, 140, 196, 265, 360, 480, 641, 842, 1104, 1432, 1855, 2378, 3040, 3858, 4888, 6146, 7708, 9616, 11969, 14818, 18305, 22511, 27629, 33773, 41191, 50069, 60744, 73453, 88645, 106681

Offset: 0

Gus Wiseman, Jun 27 2019

nonn

An integer partition has unsortable prime factors if there is no permutation (c_1,...,c_k) of the parts such that the maximum prime factor of c_i is at most the minimum prime factor of c_{i+1}. For example, the partition (27,8,6) is sortable because the permutation (8,6,27) satisfies the condition.

			The a(12) = 1 through a(17) = 14 partitions:
  (6,6)  (10,3)   (6,6,2)    (6,6,3)      (10,6)         (14,3)
         (6,6,1)  (10,3,1)   (10,3,2)     (6,6,4)        (6,6,5)
                  (6,6,1,1)  (6,6,2,1)    (10,3,3)       (10,4,3)
                             (10,3,1,1)   (6,6,2,2)      (10,6,1)
                             (6,6,1,1,1)  (6,6,3,1)      (6,6,3,2)
                                          (10,3,2,1)     (6,6,4,1)
                                          (6,6,2,1,1)    (10,3,2,2)
                                          (10,3,1,1,1)   (10,3,3,1)
                                          (6,6,1,1,1,1)  (6,6,2,2,1)
                                                         (6,6,3,1,1)
                                                         (10,3,2,1,1)
                                                         (6,6,2,1,1,1)
                                                         (10,3,1,1,1,1)
                                                         (6,6,1,1,1,1,1)

Sortable integer partitions are A326333.
Unsortable set partitions are A058681.
Unsortable normal multiset partitions are A326211.
MM-numbers of unsortable multiset partitions are A326258.
Cf. A000041, A000108, A001055, A056239, A112798, A326209, A326212.

Table[Length[Select[IntegerPartitions[n],!OrderedQ[Join@@Sort[First/@FactorInteger[#]&/@#,OrderedQ[PadRight[{#1,#2}]]&]]&]],{n,0,20}]

A000041(n) = a(n) + A326333(n).

A326334 Number of sortable factorizations of n.

Original entry on oeis.org

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

Gus Wiseman, Jun 27 2019

nonn

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.

			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)

Factorizations are A001055.
Unsortable factorizations are A326291.
Sortable integer partitions are A326333.
Cf. A058681, A326211, A326212, A326237, A326258, A326332.

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}]

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

cp's OEIS Frontend

A326332 Number of integer partitions of n with unsortable prime factors.

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