OFFSET
1,3
COMMENTS
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.
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.
Conjecture: A partition of n has weighted sum divisible by n iff its reverse has weighted sum divisible by n.
EXAMPLE
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).
The a(1) = 1 through a(9) = 5 partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(111) (11111) (222) (3211) (3311) (333)
(3111) (1111111) (221111) (4221)
(222111)
(111111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Divisible[Total[Accumulate[Reverse[#]]], n]&]], {n, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 24 2023
STATUS
approved