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

0, 1, 1, 4, 5, 8, 12, 19, 25, 38, 51, 70, 93, 124, 162, 217, 279, 360, 462, 601, 750, 955, 1203, 1502, 1881, 2336, 2892, 3596, 4407, 5416, 6623, 8083, 9830, 11943, 14471, 17488, 21059, 25317, 30376, 36424, 43489, 51906, 61789, 73498, 87186, 103253, 122098

Offset

1,4

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.

Conjecture: A partition of n has weighted sum divisible by n iff its reverse has weighted sum divisible by n.

Links

Table of n, a(n) for n=1..47.

Example

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).
The a(2) = 1 through a(7) = 12 partitions:
  (11)  (21)  (22)    (32)    (33)      (43)
              (31)    (41)    (42)      (52)
              (211)   (221)   (51)      (61)
              (1111)  (311)   (321)     (322)
                      (2111)  (411)     (331)
                              (2211)    (421)
                              (21111)   (511)
                              (111111)  (2221)
                                        (4111)
                                        (22111)
                                        (31111)
                                        (211111)

Mathematica

Table[Length[Select[IntegerPartitions[n], !Divisible[Total[Accumulate[Reverse[#]]], n]&]], {n, 30}]

Crossrefs

For median instead of mean we have A322439 aerated, complement A362558.

The complement is counted by A362559.

A000041 counts integer partitions, strict A000009.

A008284/A058398/A327482 count partitions by mean.

A264034 counts partitions by weighted sum.

A304818 = weighted sum of prime indices, row-sums of A359361.

A318283 = weighted sum of reversed prime indices, row-sums of A358136.

Cf. A001227, A051293, A067538, A240219, A261079, A326622, A349156, A360068, A360069, A360241, A362051.

Sequence in context: A027975 A011980 A260163 * A061765 A242274 A297419

Adjacent sequences: A362557 A362558 A362559 * A362561 A362562 A362563

Keyword

nonn

Author

Gus Wiseman, Apr 28 2023

Status

approved