A268602 Numerator of the side lengths (legs in ascending order) of the easiest Pythagorean Triangle (with smallest hypotenuse) according to the congruent numbers A003273.

3, 20, 41, 3, 4, 5, 35, 24, 337, 780, 323, 106921, 8, 21, 65, 4, 15, 17, 3, 40, 41, 7, 12, 25, 33, 140, 4901, 80155, 41496, 905141617, 6, 8, 10, 35, 48, 337, 99, 52780, 48029801, 5, 12, 13, 720, 8897, 2566561, 17, 24, 145, 450660, 777923, 605170417321, 1700, 5301, 1646021

Offset: 1

Author

Martin Renner, Feb 08 2016

Keywords

• nonn
• frac
• tabf

Comments

Every three fractions x < y < z satisfy the Pythagorean equation x^2 + y^2 = z^2: (a(3*n-2)/A268603(3*n-2))^2 + (a(3*n-1)/A268603(3*n-1))^2 = (a(3*n)/A268603(3*n))^2.

The area A = x*y/2 of these Pythagorean triangles is a congruent number: A003273(n) = (1/2) * (a(3*n-2)/A268603(3*n-2)) * (a(3*n-1)/A268603(3*n-1)).

Examples

			The first congruent number is 5 and the associated right triangle with the side lengths x = 3/2, y = 20/3, z = 41/6 satisfies the Pythagorean equation (3/2)^2 + (20/3)^2 = (41/6)^2 and the area of this triangle equals 1/2*3/2*20/3 = 5.
		

Links

• Editors of OEIS, Discussion of A268602, Apr 22 2023.
• Eric Weisstein's World of Mathematics, Congruent Number.

Crossrefs

Cf. A003273, A268603.

Extensions

a(14) corrected on Mar 14 2020

More terms from Jinyuan Wang, Apr 22 2023