For a positive integer $n$, let $a(n)$ the smallest number $k>0$ such that $-n$ is not a quadratic residue modulo $k$.
Using CRT, we can prove that all values of $a(n)$ are prime powers, and every prime power except $2$ appears in the sequence.
See OEIS A354597 .
It's easy to prove that $a(n)\leqslant4n-1$ (take a prime factor $p$ of $4n-1$ such that $p\equiv-1\,(\text{mod }4)$, then $-n$ is not a square modulo $p$).
But according to the figure below, it seems that we can improve the previous inequality !
How to proceed ?
Thanks !