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A240090 Number of partitions of n that have integer contraharmonic mean. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A197929 A326849 A328706 * 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

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Last modified March 31 19:16 EDT 2020. Contains 333151 sequences. (Running on oeis4.)