This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A240089 #9 Nov 09 2024 02:23:25 %S A240089 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 A240089 35,108,52,159,68,217,79,308,120,389,162,529,214,717,282,979,377,1316, %U A240089 487,1703,672,2257,904,3031,1169,3919,1517,5153,1970,6769,2544 %N A240089 Number of partitions of n having integer root mean square. %C A240089 The root mean square of a partition [x(1),..,x(k)] is sqrt((x(1)^2 + ... + x(k)^2)/k). %e A240089 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 A240089 z = 15; ColumnForm[t = Map[Select[IntegerPartitions[#], IntegerQ[RootMeanSquare[#]] &] &, Range[z]]] (* shows the partitions *) %t A240089 t1 = Map[Length, t] (* A240089 *) %t A240089 ColumnForm[u = Map[Select[IntegerPartitions[#],IntegerQ[ContraharmonicMean[#]] &] &, Range[z]]] (* shows the partitions *) %t A240089 t2 = Map[Length, u] (* A240090 *) %Y A240089 Cf. A240090. %K A240089 nonn,easy %O A240089 1,2 %A A240089 _Clark Kimberling_ and _Peter J. C. Moses_, Apr 01 2014