A343344 Number of integer partitions of n that are either empty, or do not have smallest part dividing all the others, but do have greatest part divisible by all the others.

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 5, 1, 6, 4, 6, 7, 15, 6, 16, 15, 20, 17, 36, 18, 43, 36, 46, 48, 72, 45, 93, 82, 103, 88, 152, 104, 179, 158, 191, 194, 285, 202, 328, 292, 373, 348, 502, 391, 576, 519, 659, 634, 864, 665

Offset

0,18

Comments

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

Links

Table of n, a(n) for n=0..60.

Example

The a(18) = 1  through a(23) = 15 partitions (A..E = 10..14):
  633222   C43       C332      C432       C64        E72
           A522      66332     A5222      A552       F53
           C322      633332    C3222      C433       I32
           66322     6332222   663222     C3322      C443
           633322              6333222    663322     C632
           6322222             63222222   6333322    66632
                                          63322222   C3332
                                                     C4322
                                                     663332
                                                     A52222
                                                     C32222
                                                     6333332
                                                     6632222
                                                     63332222
                                                     632222222

Mathematica

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

Crossrefs

The second condition alone gives A130689.

The half-opposite versions are A130714 and A343342.

The first condition alone gives A338470.

The Heinz numbers of these partitions are 1 and A343339.

The opposite version is A343345.

The strict case is A343380.

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, A097986, A264401, A339562, A341450, A342193, A343346.

Sequence in context: A201419 A163336 A173898 * A200644 A318265 A318553

Adjacent sequences: A343341 A343342 A343343 * A343345 A343346 A343347

Keyword

nonn

Author

Gus Wiseman, Apr 15 2021

Status

approved