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A295150 Numbers that have exactly two representations as a sum of five nonnegative squares. 10
4, 5, 8, 9, 10, 11, 12, 14, 23, 24 (list; graph; refs; listen; history; text; internal format)



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.


E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.


Table of n, a(n) for n=1..10.

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.


okQ[n_] := Length[PowersRepresentations[n, 5, 2]] == 2;

Select[Range[100], okQ] (* Jean-Fran├žois Alcover, Feb 26 2019 *)


Cf. A000174, A006431, A294675.

Sequence in context: A154885 A292192 A339499 * A042956 A128217 A288671

Adjacent sequences:  A295147 A295148 A295149 * A295151 A295152 A295153




Robert Price, Nov 15 2017



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Last modified July 27 17:57 EDT 2021. Contains 346308 sequences. (Running on oeis4.)