|
|
A363525
|
|
Number of integer partitions of n with weighted sum divisible by reverse-weighted sum.
|
|
3
|
|
|
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, 65, 55, 166, 73, 132, 242, 141, 129, 476, 183, 248, 580, 487, 312, 984, 422, 868, 1345, 825, 724, 2709, 949, 1505, 2756, 2902, 1611, 4664, 2289, 4942, 5828, 4278
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
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.
|
|
LINKS
|
|
|
EXAMPLE
|
The partition (6,5,4,3,2,1,1,1,1) has weighted sum 80, reverse 160, so is counted under a(24).
The a(n) partitions for n = 1, 2, 4, 6, 9, 12, 14 (A..E = 10-14):
1 2 4 6 9 C E
11 22 33 333 66 77
1111 222 711 444 65111
111111 6111 921 73211
111111111 3333 2222222
7311 71111111
63111 11111111111111
222222
621111
111111111111
|
|
MATHEMATICA
|
Table[Length[Select[IntegerPartitions[n], Divisible[Total[Accumulate[#]], Total[Accumulate[Reverse[#]]]]&]], {n, 30}]
|
|
CROSSREFS
|
The case of equality (and reciprocal version) is A000005.
A318283 gives weighted sum of reversed prime indices, row-sums of A358136.
Cf. A000016, A008284, A067538, A222855, A222970, A358137, A359755, A362558, A362559, A362560, A363527.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|