A363947 - OEIS

%I #14 Jul 09 2023 08:33:40

%S 0,1,1,1,2,2,2,4,4,4,7,7,7,12,12,12,19,19,19,30,30,30,45,45,45,67,67,

%T 67,97,97,97,139,139,139,195,195,195,272,272,272,373,373,373,508,508,

%U 508,684,684,684,915,915,915,1212,1212,1212,1597,1597,1597,2087

%N Number of integer partitions of n with mean < 3/2.

%e The partition y = (2,2,1) has mean 5/3, which is not less than 3/2, so y is not counted under 5.

%e The a(1) = 1 through a(8) = 4 partitions:

%e   (1)  (11)  (111)  (211)   (2111)   (21111)   (22111)    (221111)

%e                     (1111)  (11111)  (111111)  (31111)    (311111)

%e                                                (211111)   (2111111)

%e                                                (1111111)  (11111111)

%t Table[Length[Select[IntegerPartitions[n],Round[Mean[#]]==1&]],{n,0,15}]

%Y The high version is A000012 (all ones).

%Y This is A000070 with each term repeated three times (see A025065 for two).

%Y These partitions have ranks A363948.

%Y The complement is counted by A364059.

%Y A008284 counts partitions by length, A058398 by mean.

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

%Y A067538 counts partitions with integer mean, strict A102627, ranks A316413.

%Y A327482 counts partitions by integer mean.

%Y A349156 counts partitions with non-integer mean, ranks A348551.

%Y Cf. A000041, A002865, A026905, A027336, A237984, A241131, A327472, A363724, A363745, A363943, A363949.

%K nonn

%O 0,5

%A _Gus Wiseman_, Jul 02 2023