%I #6 Jan 22 2023 09:15:15

%S 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,

%T 116,381,668,2,1378,2,985,956,316,2043,4328,2,511,2293,6656,2,8634,2,

%U 8062,14671,1280,2,26228,8035,15991,11614,25055,2,47201,39810,65092

%N Number of integer partitions of n with integer mean and integer median.

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

%e The a(1) = 1 through a(9) = 9 partitions:

%e 1 2 3 4 5 6 7 8 9

%e 11 111 22 11111 33 1111111 44 333

%e 31 42 53 432

%e 1111 51 62 441

%e 222 71 522

%e 321 2222 531

%e 411 3221 621

%e 111111 3311 711

%e 5111 111111111

%e 11111111

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

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

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

%Y These partitions are ranked by A360009.

%Y A000041 counts partitions, strict A000009.

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

%Y A051293 counts subsets with integer mean, median A000975.

%Y A326567/A326568 gives mean of prime indices.

%Y A326622 counts factorizations with integer mean, strict A328966.

%Y A359893/A359901/A359902 count partitions by median.

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

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 21 2023