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 A360068 #15 Jul 09 2024 20:44:10
%S A360068 1,1,0,0,1,0,0,0,2,1,0,0,6,0,0,0,6,0,7,0,1,0,0,0,0,90,0,63,0,0,0,0,11,
%T A360068 0,0,0,436,0,0,0,0,0,0,0,0,2157,0,0,240,1595,22,0,0,0,6464,0,0,0,0,0,
%U A360068 0,0,0,11628,4361,0,0,0,0,0,0,0,12927,0,0,621,0
%N A360068 Number of integer partitions of n such that the parts have the same mean as the multiplicities.
%C A360068 Note that such a partition cannot be strict for n > 1.
%C A360068 Conjecture: If n is squarefree, then a(n) = 0.
%e A360068 The n = 1, 4, 8, 9, 12, 16, 18 partitions (D=13):
%e A360068 (1) (22) (3311) (333) (322221) (4444) (444222)
%e A360068 (5111) (332211) (43222111) (444411)
%e A360068 (422211) (43321111) (552222)
%e A360068 (522111) (53221111) (555111)
%e A360068 (531111) (54211111) (771111)
%e A360068 (621111) (63211111) (822222)
%e A360068 (D11111)
%e A360068 For example, the partition (4,3,3,3,3,3,2,2,1,1) has mean 5/2, and its multiplicities (1,5,2,2) also have mean 5/2, so it is counted under a(20).
%t A360068 Table[Length[Select[IntegerPartitions[n],Mean[#]==Mean[Length/@Split[#]]&]],{n,0,30}]
%Y A360068 These partitions are ranked by A359903, for prime factors A359904.
%Y A360068 Positions of positive terms are A360070.
%Y A360068 A000041 counts partitions, strict A000009.
%Y A360068 A058398 counts partitions by mean, see also A008284, A327482.
%Y A360068 A088529/A088530 gives mean of prime signature (A124010).
%Y A360068 A326567/A326568 gives mean of prime indices (A112798).
%Y A360068 A360069 counts partitions whose multiplicities have integer mean.
%Y A360068 Cf. A067340, A067538, A082550, A240219, A316313, A327475, A349156, A359893, A359897, A359905.
%K A360068 nonn
%O A360068 0,9
%A A360068 _Gus Wiseman_, Jan 27 2023