A positive integer $n$ is called a Hilbert number if $\exists a,b,d \in \mathbb{N}$ such that $ 4ab-a-b = d n$ and $d|a b$.
I ran an algorithm checking divisors for all $0\lt a,b\le500$, and the only numbers $n\le500$ for which I did not find a solution are
$\{1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,288,289,324,336,361,400,441,484\}$
Aside from $288,336$ they are all square numbers. It might be that a wider check would exclude these numbers, as well as some (or all) of the squares, from the list.
My Question is : Which numbers are not Hilbert Numbers ?
Basic hunch tills me that square numbers are not Hilbert Numbers
Any idea how small it is, would help my a lot.
Thanks.
Update : does every prime number also Hilbert number ?