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

0,8

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.

Links

Table of n, a(n) for n=0..50.

Example

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)

Mathematica

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

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.

Sequence in context: A035546 A339406 A182714 * A338470 A262395 A198755

Adjacent sequences: A343339 A343340 A343341 * A343343 A343344 A343345

Keyword

nonn

Author

Gus Wiseman, Apr 15 2021

Status

approved