cp's OEIS Frontend

A363525 Number of integer partitions of n with weighted sum divisible by reverse-weighted sum.

%I A363525 #9 Jun 11 2023 11:18:36
%S A363525 1,2,2,3,2,4,2,4,5,5,3,10,4,7,13,10,8,29,10,18,39,20,20,70,29,40,105,
%T A363525 65,55,166,73,132,242,141,129,476,183,248,580,487,312,984,422,868,
%U A363525 1345,825,724,2709,949,1505,2756,2902,1611,4664,2289,4942,5828,4278
%N A363525 Number of integer partitions of n with weighted sum divisible by reverse-weighted sum.
%C A363525 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.
%e A363525 The partition (6,5,4,3,2,1,1,1,1) has weighted sum 80, reverse 160, so is counted under a(24).
%e A363525 The a(n) partitions for n = 1, 2, 4, 6, 9, 12, 14 (A..E = 10-14):
%e A363525   1  2   4     6       9          C             E
%e A363525      11  22    33      333        66            77
%e A363525          1111  222     711        444           65111
%e A363525                111111  6111       921           73211
%e A363525                        111111111  3333          2222222
%e A363525                                   7311          71111111
%e A363525                                   63111         11111111111111
%e A363525                                   222222
%e A363525                                   621111
%e A363525                                   111111111111
%t A363525 Table[Length[Select[IntegerPartitions[n], Divisible[Total[Accumulate[#]], Total[Accumulate[Reverse[#]]]]&]],{n,30}]
%Y A363525 The case of equality (and reciprocal version) is A000005.
%Y A363525 The strict case is A363528.
%Y A363525 A000041 counts integer partitions, strict A000009.
%Y A363525 A053632 counts compositions by weighted sum, rank statistic A029931/A359042.
%Y A363525 A264034 counts partitions by weighted sum, reverse A358194.
%Y A363525 A304818 gives weighted sum of prime indices, row-sums of A359361.
%Y A363525 A318283 gives weighted sum of reversed prime indices, row-sums of A358136.
%Y A363525 A320387 counts multisets by weighted sum, zero-based A359678.
%Y A363525 A363526 = partitions with weighted sum 3n, ranks A363530, reverse A363531.
%Y A363525 Cf. A000016, A008284, A067538, A222855, A222970, A358137, A359755, A362558, A362559, A362560, A363527.
%K A363525 nonn
%O A363525 1,2
%A A363525 _Gus Wiseman_, Jun 10 2023