A359906 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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).

LINKS

Table of n, a(n) for n=1..56.

EXAMPLE

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

MATHEMATICA

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

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.

Sequence in context: A328710 A018818 A157019 * A067538 A305982 A304102

Adjacent sequences: A359903 A359904 A359905 * A359907 A359908 A359909

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 21 2023

STATUS

approved