A359906 Number of integer partitions of n with integer mean and integer median.

1, 2, 2, 4, 2, 8, 2, 10, 9, 14, 2, 39, 2, 24, 51, 49, 2, 109, 2, 170, 144, 69, 2, 455, 194, 116, 381, 668, 2, 1378, 2, 985, 956, 316, 2043, 4328, 2, 511, 2293, 6656, 2, 8634, 2, 8062, 14671, 1280, 2, 26228, 8035, 15991, 11614, 25055, 2, 47201, 39810, 65092

Offset: 1

Author

Gus Wiseman, Jan 21 2023

Keywords

• nonn

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The a(1) = 1 through a(9) = 9 partitions:
  1  2   3    4     5      6       7        8         9
     11  111  22    11111  33      1111111  44        333
              31           42               53        432
              1111         51               62        441
                           222              71        522
                           321              2222      531
                           411              3221      621
                           111111           3311      711
                                            5111      111111111
                                            11111111
		

Crossrefs

For just integer mean we have A067538, strict A102627, ranked by A316413.

For just integer median we have A325347, strict A359907, ranked by A359908.

These partitions are ranked by A360009.

A000041 counts partitions, strict A000009.

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

A051293 counts subsets with integer mean, median A000975.

A326567/A326568 gives mean of prime indices.

A326622 counts factorizations with integer mean, strict A328966.

A359893/A359901/A359902 count partitions by median.

A360005(n)/2 gives median of prime indices.

Cf. A000016, A082550, A237984, A240219, A326669, A327475, A349156, A359894, A359897, A359905.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], IntegerQ[Mean[#]]&&IntegerQ[Median[#]]&]],{n,1,30}]