A343341 Number of integer partitions of n with no part divisible by all the others.

1, 0, 0, 0, 0, 1, 1, 4, 6, 11, 16, 28, 36, 58, 79, 111, 149, 209, 270, 368, 472, 618, 793, 1030, 1292, 1653, 2073, 2608, 3241, 4051, 4982, 6176, 7566, 9285, 11320, 13805, 16709, 20275, 24454, 29477, 35380, 42472, 50741, 60648, 72199, 85887, 101906, 120816

Offset: 0

Author

Gus Wiseman, Apr 15 2021

Keywords

• nonn

Comments

Alternative name: Number of integer partitions of n that are either empty, or have greatest part not divisible by all the others.

Examples

			The a(5) = 1 through a(10) = 16 partitions:
  (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 A130689.

The dual version is A338470.

The Heinz numbers of these partitions are A343337.

The strict case is A343377.

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, A098965, A341450, A343342, A343345, A343346, A343381, A343382.

Programs

Mathematica

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