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

1,2

Comments

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

Links

Table of n, a(n) for n=1..59.

Example

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.

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. A240090.

Sequence in context: A280226 A307995 A061889 * A218700 A325331 A333108

Adjacent sequences: A240086 A240087 A240088 * A240090 A240091 A240092

Keyword

nonn,easy

Author

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

Status

approved