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 A294524 Numbers that have a unique partition into a sum of five nonnegative squares. 20
 0, 1, 2, 3, 6, 7, 15 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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. REFERENCES E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1. LINKS H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014 D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481. MATHEMATICA 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 *) CROSSREFS Cf. A000174, A006431, A294675. Sequence in context: A191614 A018379 A245479 * A032882 A265394 A125167 Adjacent sequences:  A294521 A294522 A294523 * A294525 A294526 A294527 KEYWORD nonn,fini,full AUTHOR Robert Price, Nov 01 2017 STATUS approved

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Last modified July 26 10:18 EDT 2021. Contains 346294 sequences. (Running on oeis4.)