  The sequence $3, 5, 9, 11, 15, 19, 21, 25, 29, 35,\dots$ consists of odd legs
in right triangles with integer side lengths and prime hypotenuse. We show that
the upper density of this sequence is zero, with logarithmic decay. The same
estimate holds for the sequence of even legs in such triangles. We expect our
upper bound, which involves the Erd\H{o}s--Ford--Tenenbaum constant, to be
sharp up to a double-logarithmic factor. We also provide a nontrivial lower
bound. Our techniques involve sieve methods, the distribution of Gaussian
primes in narrow sectors, and the Hardy--Ramanujan inequality.
