A240090 Number of partitions of n that have integer contraharmonic mean.

1, 2, 2, 3, 2, 6, 3, 7, 5, 8, 5, 17, 8, 21, 14, 31, 18, 49, 28, 56, 42, 90, 52, 146, 77, 189, 118, 257, 158, 370, 219, 530, 313, 724, 412, 999, 578, 1372, 809, 1837, 1094, 2515, 1472, 3387, 1948, 4584, 2656, 6145, 3527, 8114, 4665, 10784, 6225, 14196, 8150

Offset

1,2

Comments

The contraharmonic mean of a set {x(1),..,x(k)} is defined as (x(1)^2 + ... + x(k)^2)/(x(1) + ... + x(k)); if the set is a partition of n, this mean is (x(1)^2 + ... + x(k)^2)/n, which is the square of the root mean square of the partition, discussed at A240090.

Links

Table of n, a(n) for n=1..55.

Example

a(10) counts these 8 partitions: [10], [6,1,1,1,1], [5,5], [5,1,1,1,1,1], [4,3,2,1], [3,2,2,1,1,1], [2,2,2,2,2], [1,1,1,1,1,1,1,1,1,1]; e.g., [4,3,2,1] has contraharmonic mean (16 + 9 + 4 + 1)/10 = 3.

Mathematica

z = 15; ColumnForm[t = Map[Select[IntegerPartitions[#],      IntegerQ[RootMeanSquare[#]] &] &, Range[z]]] (* shows the partitions *)
t1 = Map[Length, t]  (* A240089 *)
ColumnForm[u = Map[Select[IntegerPartitions[#], IntegerQ[ContraharmonicMean[#]] &] &, Range[z]]] (* shows the partitions *)
t2 = Map[Length, u]  (* A240090 *)

Crossrefs

Cf. A240089.

Sequence in context: A326849 A328706 A342530 * A078224 A159688 A128710

Adjacent sequences: A240087 A240088 A240089 * A240091 A240092 A240093

Keyword

nonn,easy

Author

Clark Kimberling and Peter J. C. Moses, Apr 01 2014

Status

approved