A360069 Number of integer partitions of n whose multiset of multiplicities has integer mean.

0, 1, 2, 3, 4, 5, 9, 9, 13, 16, 25, 26, 39, 42, 62, 67, 95, 107, 147, 168, 225, 245, 327, 381, 471, 565, 703, 823, 1038, 1208, 1443, 1743, 2088, 2439, 2937, 3476, 4163, 4921, 5799, 6825, 8109, 9527, 11143, 13122, 15402, 17887, 20995, 24506, 28546, 33234, 38661

Offset: 0

Author

Gus Wiseman, Jan 27 2023

Keywords

• nonn

Examples

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (2111)   (51)      (61)       (62)
                            (11111)  (222)     (421)      (71)
                                     (321)     (2221)     (431)
                                     (2211)    (4111)     (521)
                                     (3111)    (211111)   (2222)
                                     (111111)  (1111111)  (3311)
                                                          (5111)
                                                          (221111)
                                                          (311111)
                                                          (11111111)
For example,  the partition (3,2,1,1,1,1) has multiplicities (1,1,4) with mean 2, so is counted under a(9). On the other hand, the partition (3,2,2,1,1) has multiplicities (1,2,2) with mean 5/3, so is not counted under a(9).
		

Crossrefs

These partitions are ranked by A067340 (prime signature has integer mean).

Parts instead of multiplicities: A067538, strict A102627, ranked by A316413.

The case where the parts have integer mean also is ranked by A359905.

A000041 counts integer partitions, strict A000009.

A051293 counts subsets with integer mean, median A000975.

A058398 counts partitions by mean, see also A008284, A327482.

A088529/A088530 gives mean of prime signature (A124010).

A326622 counts factorizations with integer mean, strict A328966.

Cf. A082550, A240219, A316313, A325347, A326669, A327475, A349156, A360068.

Programs

Mathematica

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