A326333 Number of integer partitions of n with sortable prime factors.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 76, 99, 132, 171, 222, 283, 363, 457, 577, 721, 902, 1115, 1379, 1693, 2076, 2530, 3077, 3723, 4500, 5410, 6494, 7765, 9270, 11025, 13089, 15491, 18307, 21569, 25369, 29765, 34869, 40750, 47546, 55361, 64367, 74685, 86529

Offset: 0

Author

Gus Wiseman, Jun 27 2019

Keywords

• nonn

Comments

An integer partition has sortable prime factors if there is a 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.

Crossrefs

Unsortable integer partitions are A326332.

Sortable normal multiset partitions are A326212.

Sortable factorizations are A326334.

Cf. A000041, A000108, A001055, A058681, A112798, A326209, A326211, A326258.

Programs

Mathematica

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

Formula

A000041(n) = a(n) + A326332(n).