cp's OEIS Frontend

A360069 Number of integer partitions of n whose multiset of multiplicities has integer mean.

%I A360069 #8 Jan 29 2023 10:45:06
%S A360069 0,1,2,3,4,5,9,9,13,16,25,26,39,42,62,67,95,107,147,168,225,245,327,
%T A360069 381,471,565,703,823,1038,1208,1443,1743,2088,2439,2937,3476,4163,
%U A360069 4921,5799,6825,8109,9527,11143,13122,15402,17887,20995,24506,28546,33234,38661
%N A360069 Number of integer partitions of n whose multiset of multiplicities has integer mean.
%e A360069 The a(1) = 1 through a(8) = 13 partitions:
%e A360069   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A360069        (11)  (21)   (22)    (32)     (33)      (43)       (44)
%e A360069              (111)  (31)    (41)     (42)      (52)       (53)
%e A360069                     (1111)  (2111)   (51)      (61)       (62)
%e A360069                             (11111)  (222)     (421)      (71)
%e A360069                                      (321)     (2221)     (431)
%e A360069                                      (2211)    (4111)     (521)
%e A360069                                      (3111)    (211111)   (2222)
%e A360069                                      (111111)  (1111111)  (3311)
%e A360069                                                           (5111)
%e A360069                                                           (221111)
%e A360069                                                           (311111)
%e A360069                                                           (11111111)
%e A360069 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 A360069 Table[Length[Select[IntegerPartitions[n], IntegerQ[Mean[Length/@Split[#]]]&]],{n,0,30}]
%Y A360069 These partitions are ranked by A067340 (prime signature has integer mean).
%Y A360069 Parts instead of multiplicities: A067538, strict A102627, ranked by A316413.
%Y A360069 The case where the parts have integer mean also is ranked by A359905.
%Y A360069 A000041 counts integer partitions, strict A000009.
%Y A360069 A051293 counts subsets with integer mean, median A000975.
%Y A360069 A058398 counts partitions by mean, see also A008284, A327482.
%Y A360069 A088529/A088530 gives mean of prime signature (A124010).
%Y A360069 A326622 counts factorizations with integer mean, strict A328966.
%Y A360069 Cf. A082550, A240219, A316313, A325347, A326669, A327475, A349156, A360068.
%K A360069 nonn
%O A360069 0,3
%A A360069 _Gus Wiseman_, Jan 27 2023