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A240089
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Number of partitions of n having integer root mean square.
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3
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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
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OFFSET
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1,2
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COMMENTS
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The root mean square of a partition [x(1),..,x(k)] is sqrt((x(1)^2 + ... + x(k)^2)/k).
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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z = 15; ColumnForm[t = Map[Select[IntegerPartitions[#], IntegerQ[RootMeanSquare[#]] &] &, Range[z]]] (* shows the partitions *)
ColumnForm[u = Map[Select[IntegerPartitions[#], IntegerQ[ContraharmonicMean[#]] &] &, Range[z]]] (* shows the partitions *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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