cp's OEIS Frontend

A362559 Number of integer partitions of n whose weighted sum is divisible by n.

%I A362559 #18 Apr 29 2023 14:39:34
%S A362559 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 A362559 118,122,158,188,219,266,313,367,412,489,578,698,809,914,1094,1268,
%U A362559 1472,1677,1948,2305,2656,3072,3527,4081,4665,5342,6225,7119,8150,9408
%N A362559 Number of integer partitions of n whose weighted sum is divisible by n.
%C A362559 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 A362559 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 A362559 Conjecture: A partition of n has weighted sum divisible by n iff its reverse has weighted sum divisible by n.
%e A362559 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 A362559 The a(1) = 1 through a(9) = 5 partitions:
%e A362559   (1)  (2)  (3)    (4)  (5)      (6)     (7)        (8)       (9)
%e A362559             (111)       (11111)  (222)   (3211)     (3311)    (333)
%e A362559                                  (3111)  (1111111)  (221111)  (4221)
%e A362559                                                               (222111)
%e A362559                                                               (111111111)
%t A362559 Table[Length[Select[IntegerPartitions[n], Divisible[Total[Accumulate[Reverse[#]]],n]&]],{n,30}]
%Y A362559 For median instead of mean we have A362558.
%Y A362559 The complement is counted by A362560.
%Y A362559 A000041 counts integer partitions, strict A000009.
%Y A362559 A008284/A058398/A327482 count partitions by mean.
%Y A362559 A264034 counts partitions by weighted sum.
%Y A362559 A304818 = weighted sum of prime indices, row-sums of A359361.
%Y A362559 A318283 = weighted sum of reversed prime indices, row-sums of A358136.
%Y A362559 Cf. A001227, A051293, A067538, A067539, A240219, A261079, A322439, A326622, A359893, A360068, A360069, A362051.
%K A362559 nonn
%O A362559 1,3
%A A362559 _Gus Wiseman_, Apr 24 2023