A360241 - OEIS

%I #8 Feb 05 2023 23:07:05

%S 0,1,2,2,4,3,8,6,13,13,22,19,43,34,56,66,97,92,156,143,233,256,322,

%T 341,555,542,710,831,1098,1131,1644,1660,2275,2484,3035,3492,4731,

%U 4848,6063,6893,8943,9378,12222,13025,16520,18748,22048,24405,31446,33698,41558

%N Number of integer partitions of n whose distinct parts have integer mean.

%e The a(1) = 1 through a(8) = 13 partitions:

%e   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)

%e        (11)  (111)  (22)    (311)    (33)      (331)      (44)

%e                     (31)    (11111)  (42)      (511)      (53)

%e                     (1111)           (51)      (3211)     (62)

%e                                      (222)     (31111)    (71)

%e                                      (321)     (1111111)  (422)

%e                                      (3111)               (2222)

%e                                      (111111)             (3221)

%e                                                           (3311)

%e                                                           (5111)

%e                                                           (32111)

%e                                                           (311111)

%e                                                           (11111111)

%e For example, the partition (32111) has distinct parts {1,2,3} with mean 2, so is counted under a(8).

%t Table[Length[Select[IntegerPartitions[n],IntegerQ[Mean[Union[#]]]&]],{n,0,30}]

%Y For parts instead of distinct parts we have A067538, ranked by A316413.

%Y The strict case is A102627.

%Y These partitions are ranked by A326621.

%Y For multiplicities instead of distinct parts: A360069, ranked by A067340.

%Y A000041 counts integer partitions, strict A000009.

%Y A008284 counts partitions by number of parts.

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

%Y A058398 counts partitions by mean, also A327482.

%Y A116608 counts partitions by number of distinct parts.

%Y A326619/A326620 gives mean of distinct prime indices.

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

%Y A360071 counts partitions by number of parts and number of distinct parts.

%Y The following count partitions:

%Y - A360242 mean(parts) != mean(distinct parts), ranked by A360246.

%Y - A360243 mean(parts) = mean(distinct parts), ranked by A360247.

%Y - A360250 mean(parts) > mean(distinct parts), ranked by A360252.

%Y - A360251 mean(parts) < mean(distinct parts), ranked by A360253.

%Y Cf. A067539, A078174, A082550, A240219, A316313, A325347, A326567/A326568, A326669, A327475, A349156.

%K nonn

%O 0,3

%A _Gus Wiseman_, Feb 02 2023