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A240089 Number of partitions of n having integer root mean square. 2

%I

%S 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,

%T 35,108,52,159,68,217,79,308,120,389,162,529,214,717,282,979,377,1316,

%U 487,1703,672,2257,904,3031,1169,3919,1517,5153,1970,6769,2544

%N Number of partitions of n having integer root mean square.

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

%e 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.

%t z = 15; ColumnForm[t = Map[Select[IntegerPartitions[#], IntegerQ[RootMeanSquare[#]] &] &, Range[z]]] (* shows the partitions *)

%t t1 = Map[Length, t] (* A240089 *)

%t ColumnForm[u = Map[Select[IntegerPartitions[#],IntegerQ[ContraharmonicMean[#]] &] &, Range[z]]] (* shows the partitions *)

%t t2 = Map[Length, u] (* A240090 *)

%Y Cf. A240090.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_ and _Peter J. C. Moses_, Apr 01 2014

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Last modified May 31 07:14 EDT 2020. Contains 334747 sequences. (Running on oeis4.)