Number that can't be root of -1 quadratic residue modulo p

Question

Every root of $−1$ quadratic residue modulo $p$ prime, $p=1(\mod4)$ is distinct.

Running tests it appears that some values are never root of $−1$ quadratic residue modulo $p$.

For exemple : $7, 18, 21, 38, 41$ etc.

Is there a way to "predict" which number will never be a root ?

Best regards.

Answer 1

The question seems to be:

Which $n$ are not square roots of $-1$ mod $p$ for some prime $p > n$ ?

This is the same as

Which $n$ have the property that the largest prime factor of $n^2+1$ is less than $n$ ?

The sequence of such $n$ is listed at OEIS as A256011. Nothing much seems to be known about it.