%I #18 Apr 29 2023 14:39:34
%S 1,1,2,1,2,3,3,3,5,4,5,7,8,11,14,14,18,25,28,26,42,47,52,73,77,100,
%T 118,122,158,188,219,266,313,367,412,489,578,698,809,914,1094,1268,
%U 1472,1677,1948,2305,2656,3072,3527,4081,4665,5342,6225,7119,8150,9408
%N Number of integer partitions of n whose weighted sum is divisible by n.
%C The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. This is also the sum of partial sums of the reverse.
%C Also the number of n-multisets of positive integers that (1) have integer mean, (2) cover an initial interval, and (3) have weakly decreasing multiplicities.
%C Conjecture: A partition of n has weighted sum divisible by n iff its reverse has weighted sum divisible by n.
%e The weighted sum of y = (4,2,2,1) is 1*4+2*2+3*2+4*1 = 18, which is a multiple of 9, so y is counted under a(9).
%e The a(1) = 1 through a(9) = 5 partitions:
%e (1) (2) (3) (4) (5) (6) (7) (8) (9)
%e (111) (11111) (222) (3211) (3311) (333)
%e (3111) (1111111) (221111) (4221)
%e (222111)
%e (111111111)
%t Table[Length[Select[IntegerPartitions[n], Divisible[Total[Accumulate[Reverse[#]]],n]&]],{n,30}]
%Y For median instead of mean we have A362558.
%Y The complement is counted by A362560.
%Y A000041 counts integer partitions, strict A000009.
%Y A008284/A058398/A327482 count partitions by mean.
%Y A264034 counts partitions by weighted sum.
%Y A304818 = weighted sum of prime indices, row-sums of A359361.
%Y A318283 = weighted sum of reversed prime indices, row-sums of A358136.
%Y Cf. A001227, A051293, A067538, A067539, A240219, A261079, A322439, A326622, A359893, A360068, A360069, A362051.
%K nonn
%O 1,3
%A _Gus Wiseman_, Apr 24 2023
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