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%I #8 Jan 29 2023 10:45:06
%S 0,1,2,3,4,5,9,9,13,16,25,26,39,42,62,67,95,107,147,168,225,245,327,
%T 381,471,565,703,823,1038,1208,1443,1743,2088,2439,2937,3476,4163,
%U 4921,5799,6825,8109,9527,11143,13122,15402,17887,20995,24506,28546,33234,38661
%N Number of integer partitions of n whose multiset of multiplicities has integer mean.
%e The a(1) = 1 through a(8) = 13 partitions:
%e (1) (2) (3) (4) (5) (6) (7) (8)
%e (11) (21) (22) (32) (33) (43) (44)
%e (111) (31) (41) (42) (52) (53)
%e (1111) (2111) (51) (61) (62)
%e (11111) (222) (421) (71)
%e (321) (2221) (431)
%e (2211) (4111) (521)
%e (3111) (211111) (2222)
%e (111111) (1111111) (3311)
%e (5111)
%e (221111)
%e (311111)
%e (11111111)
%e 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).
%t Table[Length[Select[IntegerPartitions[n], IntegerQ[Mean[Length/@Split[#]]]&]],{n,0,30}]
%Y These partitions are ranked by A067340 (prime signature has integer mean).
%Y Parts instead of multiplicities: A067538, strict A102627, ranked by A316413.
%Y The case where the parts have integer mean also is ranked by A359905.
%Y A000041 counts integer partitions, strict A000009.
%Y A051293 counts subsets with integer mean, median A000975.
%Y A058398 counts partitions by mean, see also A008284, A327482.
%Y A088529/A088530 gives mean of prime signature (A124010).
%Y A326622 counts factorizations with integer mean, strict A328966.
%Y Cf. A082550, A240219, A316313, A325347, A326669, A327475, A349156, A360068.
%K nonn
%O 0,3
%A _Gus Wiseman_, Jan 27 2023