A343342 Number of integer partitions of n with no part dividing or divisible by all the others.

1, 0, 0, 0, 0, 1, 0, 3, 2, 5, 5, 12, 7, 22, 20, 32, 34, 60, 54, 98, 93, 145, 159, 237, 229, 361, 384, 529, 574, 810, 840, 1194, 1275, 1703, 1886, 2484, 2660, 3566, 3909, 4987, 5520, 7092, 7737, 9907, 10917, 13603, 15226, 18910, 20801, 25912, 28797

Offset: 0

Author

Gus Wiseman, Apr 15 2021

Keywords

• nonn

Comments

Alternative name: Number of integer partitions of n that are either empty or have smallest part not dividing all the others and greatest part not divisible by all the others.

Examples

			The a(0) = 1 through a(12) = 7 partitions (empty columns indicated by dots):
  ()  .  .  .  .  (32)  .  (43)   (53)   (54)    (64)    (65)     (75)
                           (52)   (332)  (72)    (73)    (74)     (543)
                           (322)         (432)   (433)   (83)     (552)
                                         (522)   (532)   (92)     (732)
                                         (3222)  (3322)  (443)    (4332)
                                                         (533)    (5322)
                                                         (542)    (33222)
                                                         (722)
                                                         (3332)
                                                         (4322)
                                                         (5222)
                                                         (32222)
		

Crossrefs

The opposite version is A130714.

The first condition alone gives A338470.

The Heinz numbers of these partitions are A343338 = A342193 /\ A343337.

The second condition alone gives A343341.

The half-opposite versions are A343344 and A343345.

The "or" instead of "and" version is A343346 (strict: A343382).

The strict case is A343379.

A000009 counts strict partitions.

A000041 counts partitions.

A000070 counts partitions with a selected part (strict: A015723).

Cf. A006128, A066186, A083710, A130689, A341450, A343343, A343377.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],#=={}||!And@@IntegerQ/@(#/Min@@#)&&!And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]