A120427 For each y >= 1 there are only finitely many values of x >= 1 such that x-y and x+y are both squares; list all such pairs (x,y) with gcd(x,y) = 1 ordered by values of y; sequence gives y values.

4, 8, 12, 12, 16, 20, 20, 24, 24, 28, 28, 32, 36, 36, 40, 40, 44, 44, 48, 48, 52, 52, 56, 56, 60, 60, 60, 60, 64, 68, 68, 72, 72, 76, 76, 80, 80, 84, 84, 84, 84, 88, 88, 92, 92, 96, 96, 100, 100, 104, 104, 108, 108, 112, 112, 116, 116, 120, 120, 120, 120, 124, 124, 128

Offset: 1

Author

N. J. A. Sloane, May 02 2001

Keywords

• nonn

Comments

Ordered even legs of primitive Pythagorean triangles.

I wrote an arithmetic program once to find out if and when y 'catches up to' n in A120427 (ordered even legs of primitive Pythagorean triples). It's around 16700. As enumerated by the even - or odd - legs, (not sure about the hypotenuses), the triples are 'denser' than the integers. - Stephen Waldman, Jun 12 2007

Conjecture: lim_{n->oo} a(n)/n = 1/Pi. - Lothar Selle, Jun 19 2022

Examples

			Pairs are [5, 4], [17, 8], [13, 12], [37, 12], [65, 16], [29, 20], [101, 20], ... E.g., 5-4 = 1^2, 5+4 = 3^2.
		

References

• Lothar Selle, Kleines Handbuch Pythagoreische Zahlentripel, Books on Demand, 3rd impression 2022, chapter 2.3.1.
• Donald D. Spencer, Computers in Number Theory, Computer Science Press, Rockville MD, 1982, pp. 130-131.

Links

• D. N. Lehmer, Asymptotic evaluation of certain totient sums, Amer. J. Math. 22, 293-335, 1900.

Crossrefs

Cf. A060829, A061408, A061409.

Even entries of A024355. Ordered union of A081925 and A081935.

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, r odd, gcd(r, k) = 1.

a(n) = 2*A020887(n) = 4*A020888(n).

Extensions

Corrected by Lekraj Beedassy, Jul 12 2007 and by Stephen Waldman (brogine(AT)gmail.com), Jun 09 2007