A295489 -id:A295489 - cp's OEIS frontend

A085625 Numbers that are the sum of 2 squares in exactly 2 ways.

25, 50, 65, 85, 100, 125, 130, 145, 169, 170, 185, 200, 205, 221, 225, 250, 260, 265, 289, 290, 305, 338, 340, 365, 370, 377, 400, 410, 442, 445, 450, 481, 485, 493, 500, 505, 520, 530, 533, 545, 565, 578, 580, 585, 610, 629, 676, 680, 685, 689, 697, 730

Offset: 1

Hugo Pfoertner, Jul 09 2003

nonn

Wells erroneously writes that this sequence begins as 50, 65, 85, 145, ... . - Stefano Spezia, Sep 07 2024

			a(3) = 65 because 65 = 8^2 + 1^2 = 7^2 + 4^2;
a(4) = 85 because 85 = 9^2 + 2^2 = 7^2 + 6^2.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 125.

Robert Price, Table of n, a(n) for n = 1..16794

Cf. A000161, A000446, A224443, A294282, A295153, A295489, A295748.

Select[Range[730], Length[PowersRepresentations[#,2,2]]==2 &] (* Stefano Spezia, Sep 07 2024 *)

n such that A000161(n) = 2.

A295747 Numbers that have exactly six representations of a sum of seven nonnegative squares.

Original entry on oeis.org

21, 22, 26, 27, 32

Offset: 1

Robert Price, Nov 26 2017

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 and allows two more 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.

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.

Cf. A025422, A295489.

cp's OEIS Frontend

A085625 Numbers that are the sum of 2 squares in exactly 2 ways.

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