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

0,3

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.

Links

Table of n, a(n) for n=0..48.

Formula

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

Mathematica

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

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.

Sequence in context: A008634 A347577 A238869 * A036011 A325856 A104501

Adjacent sequences: A326330 A326331 A326332 * A326334 A326335 A326336

Keyword

nonn

Author

Gus Wiseman, Jun 27 2019

Status

approved