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A294524
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Numbers that have a unique partition into a sum of five nonnegative squares.
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20
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OFFSET
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1,3
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COMMENTS
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This sequence is finite and complete. See the von Eitzen Link and the proof in A294675 stating that for n > 5408, the number of ways to write n as a sum of 5 squares (without allowing zero squares) is at least floor(sqrt(n - 101) / 8) = 9. Since this sequence relaxes the restriction of zero squares the number of representations for n > 5408 is at least nine. Then an inspection of n <= 5408 completes the proof.
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REFERENCES
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E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.
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LINKS
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MATHEMATICA
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m = 5;
r[n_] := Reduce[xx = Array[x, m]; 0 <= x[1] && LessEqual @@ xx && AllTrue[xx, NonNegative] && n == Total[xx^2], xx, Integers];
For[n = 0, n < 20, n++, rn = r[n]; If[rn[[0]] === And, Print[n, " ", rn]]] (* Jean-François Alcover, Feb 25 2019 *)
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CROSSREFS
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KEYWORD
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nonn,fini,full
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AUTHOR
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STATUS
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approved
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