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

0,3

Links

Table of n, a(n) for n=0..50.

Example

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).

Mathematica

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

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.

Sequence in context: A357581 A174225 A182945 * A052270 A265335 A179223

Adjacent sequences: A360066 A360067 A360068 * A360070 A360071 A360072

Keyword

nonn

Author

Gus Wiseman, Jan 27 2023

Status

approved