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Number of integer partitions of n whose weighted sum is not divisible by n.
7

%I #6 Apr 29 2023 14:11:45

%S 0,1,1,4,5,8,12,19,25,38,51,70,93,124,162,217,279,360,462,601,750,955,

%T 1203,1502,1881,2336,2892,3596,4407,5416,6623,8083,9830,11943,14471,

%U 17488,21059,25317,30376,36424,43489,51906,61789,73498,87186,103253,122098

%N Number of integer partitions of n whose weighted sum is not 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 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 = (3,3,1) is 1*3+2*3+3*1 = 12, which is not a multiple of 7, so y is counted under a(7).

%e The a(2) = 1 through a(7) = 12 partitions:

%e (11) (21) (22) (32) (33) (43)

%e (31) (41) (42) (52)

%e (211) (221) (51) (61)

%e (1111) (311) (321) (322)

%e (2111) (411) (331)

%e (2211) (421)

%e (21111) (511)

%e (111111) (2221)

%e (4111)

%e (22111)

%e (31111)

%e (211111)

%t Table[Length[Select[IntegerPartitions[n],!Divisible[Total[Accumulate[Reverse[#]]],n]&]],{n,30}]

%Y For median instead of mean we have A322439 aerated, complement A362558.

%Y The complement is counted by A362559.

%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, A240219, A261079, A326622, A349156, A360068, A360069, A360241, A362051.

%K nonn

%O 1,4

%A _Gus Wiseman_, Apr 28 2023