A295150 Numbers that have exactly two representations as a sum of five nonnegative squares.

4, 5, 8, 9, 10, 11, 12, 14, 23, 24

Offset: 1

Author

Robert Price, Nov 15 2017

Keywords

• nonn
• fini
• full

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.

Crossrefs

Cf. A000174, A006431, A294675.

Programs

Mathematica

okQ[n_] := Length[PowersRepresentations[n, 5, 2]] == 2;
Select[Range[100], okQ] (* Jean-François Alcover, Feb 26 2019 *)