A120427 - OEIS

%I #28 Jun 20 2022 12:18:19

%S 4,8,12,12,16,20,20,24,24,28,28,32,36,36,40,40,44,44,48,48,52,52,56,

%T 56,60,60,60,60,64,68,68,72,72,76,76,80,80,84,84,84,84,88,88,92,92,96,

%U 96,100,100,104,104,108,108,112,112,116,116,120,120,120,120,124,124,128

%N 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.

%C Ordered even legs of primitive Pythagorean triangles.

%C 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

%C Conjecture: lim_{n->oo} a(n)/n = 1/Pi. - _Lothar Selle_, Jun 19 2022

%D Lothar Selle, Kleines Handbuch Pythagoreische Zahlentripel, Books on Demand, 3rd impression 2022, chapter 2.3.1.

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

%H D. N. Lehmer, <a href="http://www.jstor.org/stable/2369728">Asymptotic evaluation of certain totient sums</a>, Amer. J. Math. 22, 293-335, 1900.

%F 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.

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

%e 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.

%Y Cf. A060829, A061408, A061409.

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

%K nonn

%O 1,1

%A _N. J. A. Sloane_, May 02 2001

%E Corrected by _Lekraj Beedassy_, Jul 12 2007 and by Stephen Waldman (brogine(AT)gmail.com), Jun 09 2007