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 A360690 #6 Feb 24 2023 21:46:46
%S A360690 0,0,0,1,2,2,5,6,8,8,14,12,21,20,31,36,57,61,94,108,157,188,261,305,
%T A360690 409,484,632,721,942,1083,1376,1585,2004,2302,2860,3304,4103,4742,
%U A360690 5849,6745,8281,9599,11706,13605,16481,19176,23078,26838,32145,37387,44465
%N A360690 Number of integer partitions of n with non-integer median of multiplicities.
%C A360690 The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
%e A360690 The a(1) = 0 through a(9) = 8 partitions:
%e A360690 . . . (211) (221) (411) (322) (332) (441)
%e A360690 (311) (21111) (331) (422) (522)
%e A360690 (511) (611) (711)
%e A360690 (22111) (22211) (22221)
%e A360690 (31111) (41111) (33111)
%e A360690 (2111111) (51111)
%e A360690 (2211111)
%e A360690 (3111111)
%e A360690 For example, the partition y = (3,2,2,1) has multiplicities (1,2,1), and the multiset {1,1,2} has median 1, so y is not counted under a(8).
%t A360690 Table[Length[Select[IntegerPartitions[n], !IntegerQ[Median[Length/@Split[#]]]&]],{n,30}]
%Y A360690 These partitions have ranks A360554.
%Y A360690 The complement is counted by A360687, ranks A360553.
%Y A360690 A058398 counts partitions by mean, see also A008284, A327482.
%Y A360690 A124010 gives prime signature, sorted A118914, mean A088529/A088530.
%Y A360690 A325347 = partitions w/ integer median, strict A359907, complement A307683.
%Y A360690 A359893 and A359901 count partitions by median, odd-length A359902.
%Y A360690 A360069 = partitions with integer mean of multiplicities, ranks A067340.
%Y A360690 Cf. A000041, A000975, A090794, A329976, A359908, A360068, A360460, A360550, A360556, A360688.
%K A360690 nonn
%O A360690 1,5
%A A360690 _Gus Wiseman_, Feb 22 2023