A268603 - OEIS

%I #23 Feb 16 2025 08:33:30

%S 2,3,6,1,1,1,12,5,60,323,30,9690,3,2,6,1,2,2,1,3,3,2,1,2,35,3,105,

%T 20748,3485,72306780

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

%C Every three fractions x < y < z satisfy the Pythagorean equation x^2 + y^2 = z^2: (A268602(3*n-2)/a(3*n-2))^2 + (A268602(3*n-1)/a(3*n-1))^2 = (A268602(3*n)/a(3*n))^2.

%C The area A = x*y/2 of these Pythagorean triangles is a congruent number: A003273(n) = (1/2) * A268602(3*n-2)/a(3*n-2) * A268602(3*n-1)/a(3*n-1)).

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CongruentNumber.html">Congruent Number.</a>

%e 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.

%Y Cf. A003273, A268602.

%K nonn,frac,tabf,more,changed

%O 1,1

%A _Martin Renner_, Feb 08 2016

%E a(14) corrected on Mar 14 2020