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A080673
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Largest number with exactly n representations as sum of five positive squares (or 0 if no number with exactly n representations exists).
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4
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33, 60, 105, 90, 132, 177, 145, 201, 225, 180, 297, 228, 213, 265, 345, 258, 305, 300, 393, 369, 465, 417, 385, 337, 420, 425, 513, 537, 409, 473, 449, 585, 561, 545, 481, 505, 633, 540, 528, 705, 593, 462, 665, 681, 617, 825, 564, 753, 548, 689, 777, 713
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OFFSET
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0,1
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COMMENTS
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a(0) = 33 has been asked as a riddle by Gerhard Woeginger in de.rec.denksport.
There is no number <= 10^6 that is the sum of five positive squares in exactly 188 ways. - Donovan Johnson, Aug 15 2013
Donovan Johnson's exhaustive search in fact shows that a(188) is undefined / should be assigned the default value 0. Hagen von Eitzen, Jun 05 2014
In what sense the search is exhaustive? Is any of the 0's of the b-file proved? - M. F. Hasler, Oct 27 2017
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LINKS
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FORMULA
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EXAMPLE
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A known result says a(0) = 33, since there is no representation of 33 as sum of 5 positive squares.
a(1) = 60 is the largest number allowing exactly one such representation.
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MATHEMATICA
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max = 1000; m = Ceiling[Sqrt[max]]; xx = Array[x, 6, 0]; x[0] = 1; iter = Sequence @@ Thread[{Rest[xx], Most[xx], m}]; representations = Table[ Rest[xx] . Rest[xx], Evaluate[iter]] // Flatten // Sort // Split // Select[#, First[#] <= max &] &; counts = {First[#], Length[#]} & /@ representations; a[0] = Complement[Range[max], counts[[All, 1]]] // Last; a[n_] := Select[counts, #[[2]] == n &] // Last // First; Table[a[n], {n, 0, 51}] (* Jean-François Alcover, Jul 12 2012 *)
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CROSSREFS
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KEYWORD
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easy,nice,nonn
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AUTHOR
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EXTENSIONS
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Definition adjusted to cope with otherwise undefined values and b-file extended by Hagen von Eitzen, Jun 04 2014
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STATUS
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approved
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