A343346 Number of integer partitions of n that are empty, have smallest part not dividing all the others, or greatest part not divisible by all the others.

1, 0, 0, 0, 0, 1, 1, 4, 6, 11, 16, 29, 36, 59, 80, 112, 150, 214, 271, 374, 476, 624, 800, 1045, 1298, 1669, 2088, 2628, 3258, 4087, 5000, 6219, 7602, 9331, 11368, 13877, 16754, 20368, 24536, 29580, 35468, 42624, 50845, 60827, 72357, 86078, 102100, 121101

Offset: 0

Author

Gus Wiseman, Apr 15 2021

Keywords

• nonn

Comments

First differs from A343345 at a(14) = 80, A343345(14) = 79.

Alternative name: Number of integer partitions of n with either no part dividing, or no part divisible by all the others.

Examples

			The a(0) = 1 through a(10) = 16 partitions (empty columns indicated by dots):
  ()  .  .  .  .  (32)  (321)  (43)    (53)     (54)      (64)
                               (52)    (332)    (72)      (73)
                               (322)   (431)    (432)     (433)
                               (3211)  (521)    (522)     (532)
                                       (3221)   (531)     (541)
                                       (32111)  (3222)    (721)
                                                (3321)    (3322)
                                                (4311)    (4321)
                                                (5211)    (5221)
                                                (32211)   (5311)
                                                (321111)  (32221)
                                                          (33211)
                                                          (43111)
                                                          (52111)
                                                          (322111)
                                                          (3211111)
		

Crossrefs

The complement is counted by A130714.

The first condition alone gives A338470.

The second condition alone gives A343341.

The "and" instead of "or" version is A343342.

The Heinz numbers of these partitions are A343343.

The strict case is A343382.

A000009 counts strict partitions.

A000041 counts partitions.

A000070 counts partitions with a selected part.

A006128 counts partitions with a selected position.

A015723 counts strict partitions with a selected part.

Cf. A083710, A130689, A341450, A342193, A343337, A343338, A343379.

Programs

Mathematica

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