A327472 - OEIS

%I #8 Sep 22 2023 01:58:47

%S 1,0,0,1,2,5,6,13,16,25,34,54,56,99,121,154,201,295,324,488,541,725,

%T 957,1253,1292,1892,2356,2813,3378,4563,4838,6840,7686,9600,12076,

%U 14180,15445,21635,25627,29790,33309,44581,48486,63259,70699,82102,104553,124752

%N Number of integer partitions of n not containing their mean.

%e The a(3) = 1 through a(8) = 16 partitions not containing their mean:

%e   (21)  (31)   (32)    (42)     (43)      (53)

%e         (211)  (41)    (51)     (52)      (62)

%e                (221)   (411)    (61)      (71)

%e                (311)   (2211)   (322)     (332)

%e                (2111)  (3111)   (331)     (422)

%e                        (21111)  (421)     (431)

%e                                 (511)     (521)

%e                                 (2221)    (611)

%e                                 (3211)    (3311)

%e                                 (4111)    (5111)

%e                                 (22111)   (22211)

%e                                 (31111)   (32111)

%e                                 (211111)  (41111)

%e                                           (221111)

%e                                           (311111)

%e                                           (2111111)

%t Table[Length[Select[IntegerPartitions[n],!MemberQ[#,Mean[#]]&]],{n,0,20}]

%o (Python)

%o from sympy.utilities.iterables import partitions

%o def A327472(n): return sum(1 for s,p in partitions(n,size=True) if n%s or n//s not in p) if n else 1 # _Chai Wah Wu_, Sep 21 2023

%Y The Heinz numbers of these partitions are A327476.

%Y Partitions with their mean are A237984.

%Y Subsets without their mean are A327471.

%Y Subsets with n but without their mean are A327477.

%Y Strict partitions without their mean are A240851.

%Y Cf. A007865, A067538, A067538, A102627, A114639, A324756, A327473.

%K nonn

%O 0,5

%A _Gus Wiseman_, Sep 13 2019