A061409 - OEIS

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A061409

For each y >= 1 there are only finitely many values of x >= 1 such that x-y and x+y are both positive squares; list all such pairs (x,y) ordered by values of y; sequence gives x values.

5, 10, 17, 26, 13, 37, 50, 20, 65, 82, 29, 101, 122, 25, 40, 145, 170, 53, 197, 34, 226, 68, 257, 290, 45, 85, 325, 362, 41, 104, 401, 58, 442, 125, 485, 530, 52, 73, 148, 577, 626, 173, 677, 90, 730, 65, 200, 785, 842, 61, 109, 229, 901, 962

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OFFSET

0,1

REFERENCES

Donald D. Spencer, Computers in Number Theory, Computer Science Press, Rockville MD, 1982, pp. 130-131.

LINKS

Robert Israel, Table of n, a(n) for n = 0..10000

FORMULA

The solutions are given by x = r^2 + 2*r*k + 2*k^2, y = 2*k*(k+r) with r >= 1, k >= 1. - N. J. A. Sloane, May 02 2001

EXAMPLE

Pairs are [5, 4], [10, 6], [17, 8], [26, 10], [13, 12], [37, 12], [50, 14], ... For example, 5-4 = 1^2, 5+4 = 3^2.

MAPLE

seq(op(sort(map(k -> (k^2 + (y/2/k)^2), select(t -> t^2 < y/2, convert(numtheory:-divisors(y/2), list))))), y=2..100, 2); # Robert Israel, Dec 10 2017

CROSSREFS

Cf. A061408, A060829, A060830.

Sequence in context: A307028 A276382 A105705 * A342553 A229997 A277186

Adjacent sequences: A061406 A061407 A061408 * A061410 A061411 A061412

KEYWORD

nonn

AUTHOR

Jason Earls, May 01 2001

EXTENSIONS

Definition clarified by Robert Israel, Dec 10 2017

STATUS

approved