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

Author

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

Keywords

• nonn
• easy

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.

Examples

			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.
		

Crossrefs

Cf. A240089.

Programs

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 *)