This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A349156 #10 Dec 04 2021 12:37:48
%S A349156 1,0,0,1,1,5,3,13,11,21,28,54,31,99,111,125,165,295,259,488,425,648,
%T A349156 933,1253,943,1764,2320,2629,2962,4563,3897,6840,6932,9187,11994,
%U A349156 12840,12682,21635,25504,28892,28187,44581,42896,63259,66766,74463,104278,124752
%N A349156 Number of integer partitions of n whose mean is not an integer.
%C A349156 Equivalently, partitions whose length does not divide their sum.
%C A349156 By conjugation, also the number of integer partitions of n with greatest part not dividing n.
%F A349156 a(n > 0) = A000041(n) - A067538(n).
%e A349156 The a(3) = 1 through a(8) = 11 partitions:
%e A349156 (21) (211) (32) (2211) (43) (332)
%e A349156 (41) (3111) (52) (422)
%e A349156 (221) (21111) (61) (431)
%e A349156 (311) (322) (521)
%e A349156 (2111) (331) (611)
%e A349156 (421) (22211)
%e A349156 (511) (32111)
%e A349156 (2221) (41111)
%e A349156 (3211) (221111)
%e A349156 (4111) (311111)
%e A349156 (22111) (2111111)
%e A349156 (31111)
%e A349156 (211111)
%t A349156 Table[Length[Select[IntegerPartitions[n],!IntegerQ[Mean[#]]&]],{n,0,30}]
%Y A349156 Below, "!" means either enumerative or set theoretical complement.
%Y A349156 The version for nonempty subsets is !A051293.
%Y A349156 The complement is counted by A067538, ranked by A316413.
%Y A349156 The geometric version is !A067539, strict !A326625, ranked by !A326623.
%Y A349156 The strict case is !A102627.
%Y A349156 The version for prime factors is A175352, complement A078175.
%Y A349156 The version for distinct prime factors is A176587, complement A078174.
%Y A349156 The ordered version (compositions) is !A271654, ranked by !A096199.
%Y A349156 The multiplicative version (factorizations) is !A326622, geometric !A326028.
%Y A349156 The conjugate is ranked by !A326836.
%Y A349156 The conjugate strict version is !A326850.
%Y A349156 These partitions are ranked by A348551.
%Y A349156 A000041 counts integer partitions.
%Y A349156 A326567/A326568 give the mean of prime indices, conjugate A326839/A326840.
%Y A349156 A236634 counts unbalanced partitions, complement of A047993.
%Y A349156 A327472 counts partitions not containing their mean, complement of A237984.
%Y A349156 Cf. A001700, A074761, A098743, A143773, A175397, A175761, A298423, A326027, A326641, A326842, A326849, A327778.
%K A349156 nonn
%O A349156 0,6
%A A349156 _Gus Wiseman_, Nov 14 2021