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A326332
Number of integer partitions of n with unsortable prime factors.
3
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
0,14
COMMENTS
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.
FORMULA
A000041(n) = a(n) + A326333(n).
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], !OrderedQ[Join@@Sort[First/@FactorInteger[#]&/@#, OrderedQ[PadRight[{#1, #2}]]&]]&]], {n, 0, 20}]
CROSSREFS
Sortable integer partitions are A326333.
Unsortable set partitions are A058681.
Unsortable normal multiset partitions are A326211.
MM-numbers of unsortable multiset partitions are A326258.
Sequence in context: A280204 A240841 A056686 * A079962 A244986 A293547
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 27 2019
STATUS
approved