A360241 Number of integer partitions of n whose distinct parts have integer mean.

0, 1, 2, 2, 4, 3, 8, 6, 13, 13, 22, 19, 43, 34, 56, 66, 97, 92, 156, 143, 233, 256, 322, 341, 555, 542, 710, 831, 1098, 1131, 1644, 1660, 2275, 2484, 3035, 3492, 4731, 4848, 6063, 6893, 8943, 9378, 12222, 13025, 16520, 18748, 22048, 24405, 31446, 33698, 41558

Offset: 0

Author

Gus Wiseman, Feb 02 2023

Keywords

• nonn

Examples

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (331)      (44)
                    (31)    (11111)  (42)      (511)      (53)
                    (1111)           (51)      (3211)     (62)
                                     (222)     (31111)    (71)
                                     (321)     (1111111)  (422)
                                     (3111)               (2222)
                                     (111111)             (3221)
                                                          (3311)
                                                          (5111)
                                                          (32111)
                                                          (311111)
                                                          (11111111)
For example, the partition (32111) has distinct parts {1,2,3} with mean 2, so is counted under a(8).
		

Crossrefs

For parts instead of distinct parts we have A067538, ranked by A316413.

The strict case is A102627.

These partitions are ranked by A326621.

For multiplicities instead of distinct parts: A360069, ranked by A067340.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by number of parts.

A051293 counts subsets with integer mean, median A000975.

A058398 counts partitions by mean, also A327482.

A116608 counts partitions by number of distinct parts.

A326619/A326620 gives mean of distinct prime indices.

A326622 counts factorizations with integer mean, strict A328966.

A360071 counts partitions by number of parts and number of distinct parts.

The following count partitions:

- A360242 mean(parts) != mean(distinct parts), ranked by A360246.

- A360243 mean(parts) = mean(distinct parts), ranked by A360247.

- A360250 mean(parts) > mean(distinct parts), ranked by A360252.

- A360251 mean(parts) < mean(distinct parts), ranked by A360253.

Cf. A067539, A078174, A082550, A240219, A316313, A325347, A326567/A326568, A326669, A327475, A349156.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],IntegerQ[Mean[Union[#]]]&]],{n,0,30}]