cp's OEIS Frontend

A360241 Number of integer partitions of n whose distinct parts have integer mean.

%I A360241 #8 Feb 05 2023 23:07:05
%S A360241 0,1,2,2,4,3,8,6,13,13,22,19,43,34,56,66,97,92,156,143,233,256,322,
%T A360241 341,555,542,710,831,1098,1131,1644,1660,2275,2484,3035,3492,4731,
%U A360241 4848,6063,6893,8943,9378,12222,13025,16520,18748,22048,24405,31446,33698,41558
%N A360241 Number of integer partitions of n whose distinct parts have integer mean.
%e A360241 The a(1) = 1 through a(8) = 13 partitions:
%e A360241   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A360241        (11)  (111)  (22)    (311)    (33)      (331)      (44)
%e A360241                     (31)    (11111)  (42)      (511)      (53)
%e A360241                     (1111)           (51)      (3211)     (62)
%e A360241                                      (222)     (31111)    (71)
%e A360241                                      (321)     (1111111)  (422)
%e A360241                                      (3111)               (2222)
%e A360241                                      (111111)             (3221)
%e A360241                                                           (3311)
%e A360241                                                           (5111)
%e A360241                                                           (32111)
%e A360241                                                           (311111)
%e A360241                                                           (11111111)
%e A360241 For example, the partition (32111) has distinct parts {1,2,3} with mean 2, so is counted under a(8).
%t A360241 Table[Length[Select[IntegerPartitions[n],IntegerQ[Mean[Union[#]]]&]],{n,0,30}]
%Y A360241 For parts instead of distinct parts we have A067538, ranked by A316413.
%Y A360241 The strict case is A102627.
%Y A360241 These partitions are ranked by A326621.
%Y A360241 For multiplicities instead of distinct parts: A360069, ranked by A067340.
%Y A360241 A000041 counts integer partitions, strict A000009.
%Y A360241 A008284 counts partitions by number of parts.
%Y A360241 A051293 counts subsets with integer mean, median A000975.
%Y A360241 A058398 counts partitions by mean, also A327482.
%Y A360241 A116608 counts partitions by number of distinct parts.
%Y A360241 A326619/A326620 gives mean of distinct prime indices.
%Y A360241 A326622 counts factorizations with integer mean, strict A328966.
%Y A360241 A360071 counts partitions by number of parts and number of distinct parts.
%Y A360241 The following count partitions:
%Y A360241 - A360242 mean(parts) != mean(distinct parts), ranked by A360246.
%Y A360241 - A360243 mean(parts) = mean(distinct parts), ranked by A360247.
%Y A360241 - A360250 mean(parts) > mean(distinct parts), ranked by A360252.
%Y A360241 - A360251 mean(parts) < mean(distinct parts), ranked by A360253.
%Y A360241 Cf. A067539, A078174, A082550, A240219, A316313, A325347, A326567/A326568, A326669, A327475, A349156.
%K A360241 nonn
%O A360241 0,3
%A A360241 _Gus Wiseman_, Feb 02 2023