Ubiquitous Hypotenuses

You should specify that the numbers can be written as the sum of two positive square numbers.

For this problem I refered to the handy parametrization of pythagorean triples: $x=(2ab)k, y=(a^2-b^2)k, z=(a^2+b^2)k$ with $(a,b)=1,k$ integer.

In this case, $a\ne b$ due to the positive restriction. Now, we want to fix $z$ , and for it to satisfy the condition, two or more of its factors have to be able to be written as a sum of two distinct non-zero squares. We can easily list the first few possible factors:

$5,10,13,17,25,26...$

Product of two or more of these numbers will give a satisfactory $z$ (unless one of the numbers already satisfy the condition, in that case you only need one number. e.g. $25$ can stand on its own), the next smallest product not divisible by $5$ is hence $13^2=\boxed {169}$