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A240089 Number of partitions of n having integer root mean square. 2
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 (list; graph; refs; listen; history; text; internal format)
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 A266935

Adjacent sequences:  A240086 A240087 A240088 * A240090 A240091 A240092

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

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Last modified March 30 16:16 EDT 2020. Contains 333127 sequences. (Running on oeis4.)