A240089 Number of partitions of n having integer root mean square.

1, 2, 2, 3, 2, 4, 3, 6, 3, 6, 2, 9, 4, 9, 6, 17, 5, 20, 9, 19, 13, 31, 14, 47, 19, 68, 24, 90, 35, 108, 52, 159, 68, 217, 79, 308, 120, 389, 162, 529, 214, 717, 282, 979, 377, 1316, 487, 1703, 672, 2257, 904, 3031, 1169, 3919, 1517, 5153, 1970, 6769, 2544

Offset: 1

Author

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

Keywords

• nonn
• easy

Comments

The root mean square of a partition [x(1),..,x(k)] is sqrt((x(1)^2 + ... + x(k)^2)/k).

Examples

			a(10) counts these 6 partitions: [10], [5,5], [5,3,1,1], [4,2,1,1,1,1], [2,2,2,2,2], [1,1,1,1,1,1,1,1,1,1]; e.g., [5,3,1,1] has root mean square sqrt((25 + 9 + 1 + 1)/4) = 3.
		

Crossrefs

Cf. A240090.

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 *)