A294736 Numbers that are the sum of 5 nonzero squares in exactly 2 ways.

20, 38, 41, 45, 47, 48, 49, 50, 54, 55, 63, 66, 81, 105

Offset: 1

Author

Robert Price, Nov 07 2017

Keywords

• nonn
• fini
• full

Comments

Inspected values of n <= 50000.

This sequence is complete, see the von Eitzen Link and Price's computation that the next term must be > 50000. Proof. The link mentions "for positive integer n, if n > 5408 then the number of ways to write n as a sum of 5 squares is at least Floor(Sqrt(n - 101) / 8)". So for n > 5408, there are more than two ways to write n as a sum of 5 squares. For n <= 5408, it has been verified if n is in the sequence by inspection. Hence the sequence is complete." - David A. Corneth, Nov 08 2017

Examples

			There are exactly two ways 20 is a sum of 5 nonzero squares. These are 1^2 + 1^2 + 1^2 + 1^2 + 4^2 = 2^2 + 2^2 + 2^2 + 2^2 + 2^2 = 20. Therefore 20 is in the sequence.
		

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.
• Eric Weisstein's World of Mathematics, Square Number
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A025429, A025357, A243148, A294675.

Programs

Mathematica

Select[Range[200], Length[Select[PowersRepresentations[#, 5, 2], #[[1]] > 0&]] == 2&] (* Jean-François Alcover, Nov 06 2020 *)

Formula

A243148(a(n),5) = 2. - Alois P. Heinz, Feb 26 2019