A339619 Number of integer partitions of n with no 1's and a part divisible by all the other parts.

1, 0, 1, 1, 2, 1, 4, 1, 5, 3, 7, 2, 13, 2, 13, 9, 17, 6, 27, 7, 33, 19, 35, 16, 58, 22, 58, 39, 75, 37, 108, 44, 117, 75, 132, 88, 190, 94, 199, 147, 250, 153, 322, 180, 363, 271, 405, 286, 544, 339, 601, 458, 699, 503, 868, 608, 990, 777, 1113, 865, 1422, 993

Offset: 0

Author

Gus Wiseman, Apr 18 2021

Keywords

• nonn

Comments

Alternative name: Number of integer partitions of n with no 1's that are empty or have greatest part divisible by all the other parts.

Examples

			The a(6) = 4 through a(16) = 17 partitions (A..G = 10..16):
  6    7  8     9    A      B    C       D     E        F      G
  33      44    63   55     632  66      6322  77       A5     88
  42      62    333  82          84            C2       C3     C4
  222     422        442         93            662      555    E2
          2222       622         A2            842      663    844
                     4222        444           A22      933    C22
                     22222       633           4442     6333   4444
                                 822           6332     33333  6622
                                 3333          8222     63222  8422
                                 4422          44222           A222
                                 6222          62222           44422
                                 42222         422222          63322
                                 222222        2222222         82222
                                                               442222
                                                               622222
                                                               4222222
                                                               22222222
		

Crossrefs

The dual version is A083711.

The version with 1's allowed is A130689.

The strict case is A339660.

The Heinz numbers of these partitions are the odd complement of A343337.

The strict case with 1's allowed is A343347.

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. A066186, A083710, A083711, A097986, A130714, A338470, A343341, A343342, A343346, A343377, A343382.

Programs

Mathematica

Table[If[n==0,1,Length[Select[IntegerPartitions[n],FreeQ[#,1]&&Or@@And@@IntegerQ/@(Max@@#/#)&]]],{n,0,30}]