If every divisor of $x$ divides a number of the form $y^2+2$, what can be said about $x$?

Question

Let $x$ be a positive integer such that every divisor of $x$ (including $x$, of course) divides an integer of the form $y^2+2$.

By classical results, we know that every prime factor is of the form $a^2+2b^2$.

QUESTION: Does $x$ have any special properties beyond that?

Answer 1

Well, since the set $E$ of numbers of the form $a^2+2b^2$ is a semigroup due to the generalized Lagrange's identity $$ (a^2+2b^2)(c^2+2d^2) = (ac+2bd)^2 + 2(ad-bc)^2 $$ we have that $x$ itself belongs to $E$, for instance.