A343380 Number of strict integer partitions of n with no part dividing all the others but with a part divisible by all the others.

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 1, 1, 4, 0, 1, 0, 2, 0, 4, 0, 3, 1, 2, 2, 5, 0, 5, 3, 4, 1, 9, 1, 5, 2, 4, 5, 11, 1, 6, 4, 11, 3, 13, 5, 10, 4, 11, 8, 14, 3, 10, 6, 9, 3, 15, 6, 14, 10, 18, 8

Offset: 0

Author

Gus Wiseman, Apr 16 2021

Keywords

• nonn

Comments

Alternative name: Number of strict integer partitions of n that are either empty or (1) have smallest part not dividing all the others and (2) have greatest part divisible by all the others.

Examples

			The a(11) = 1 through a(29) = 4 partitions (empty columns indicated by dots, A..O = 10..24):
  632  .  .  .  .  .  A52  .  C43  .  C432  C64  E72   .  C643  .  K52    .  I92
                      C32                        F53               C6432     K54
                                                 I32                         O32
                                                 C632                        I632
		

Crossrefs

The first condition alone gives A341450.

The non-strict version is A343344 (Heinz numbers: A343339).

The second condition alone gives A343347.

The half-opposite versions are A343378 and A343379.

The opposite (and dual) version is A343381.

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.

A018818 counts partitions into divisors (strict: A033630).

A167865 counts strict chains of divisors > 1 summing to n.

A339564 counts factorizations with a selected factor.

Cf. A083710, A097986, A130689, A200745, A264401, A338470, A339562, A342193, A343377, A343382.

Programs

Mathematica

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