A240090 - OEIS

%I #6 Apr 09 2014 21:52:35

%S 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,

%T 118,257,158,370,219,530,313,724,412,999,578,1372,809,1837,1094,2515,

%U 1472,3387,1948,4584,2656,6145,3527,8114,4665,10784,6225,14196,8150

%N Number of partitions of n that have integer contraharmonic mean.

%C 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.

%e 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.

%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. A240089.

%K nonn,easy

%O 1,2

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