cp's OEIS Frontend

A362560 Number of integer partitions of n whose weighted sum is not divisible by n.

%I A362560 #6 Apr 29 2023 14:11:45
%S A362560 0,1,1,4,5,8,12,19,25,38,51,70,93,124,162,217,279,360,462,601,750,955,
%T A362560 1203,1502,1881,2336,2892,3596,4407,5416,6623,8083,9830,11943,14471,
%U A362560 17488,21059,25317,30376,36424,43489,51906,61789,73498,87186,103253,122098
%N A362560 Number of integer partitions of n whose weighted sum is not divisible by n.
%C A362560 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 A362560 Conjecture: A partition of n has weighted sum divisible by n iff its reverse has weighted sum divisible by n.
%e A362560 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 A362560 The a(2) = 1 through a(7) = 12 partitions:
%e A362560   (11)  (21)  (22)    (32)    (33)      (43)
%e A362560               (31)    (41)    (42)      (52)
%e A362560               (211)   (221)   (51)      (61)
%e A362560               (1111)  (311)   (321)     (322)
%e A362560                       (2111)  (411)     (331)
%e A362560                               (2211)    (421)
%e A362560                               (21111)   (511)
%e A362560                               (111111)  (2221)
%e A362560                                         (4111)
%e A362560                                         (22111)
%e A362560                                         (31111)
%e A362560                                         (211111)
%t A362560 Table[Length[Select[IntegerPartitions[n],!Divisible[Total[Accumulate[Reverse[#]]],n]&]],{n,30}]
%Y A362560 For median instead of mean we have A322439 aerated, complement A362558.
%Y A362560 The complement is counted by A362559.
%Y A362560 A000041 counts integer partitions, strict A000009.
%Y A362560 A008284/A058398/A327482 count partitions by mean.
%Y A362560 A264034 counts partitions by weighted sum.
%Y A362560 A304818 = weighted sum of prime indices, row-sums of A359361.
%Y A362560 A318283 = weighted sum of reversed prime indices, row-sums of A358136.
%Y A362560 Cf. A001227, A051293, A067538, A240219, A261079, A326622, A349156, A360068, A360069, A360241, A362051.
%K A362560 nonn
%O A362560 1,4
%A A362560 _Gus Wiseman_, Apr 28 2023