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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
Sequence in context: A191614 A018379 A245479 * A032882 A265394 A125167
KEYWORD
nonn,fini,full
AUTHOR
Robert Price, Nov 01 2017
STATUS
approved

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Last modified March 28 14:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)