An open and famously difficult problem is to (dis)prove the infinitude of primes of the form $n^2+1$. My question is related to these primes:
Suppose there are infinitely many primes of the form $n^2+1$. Let $p_k$ denote the $k$-th prime and $s_k$ the $k$-th prime of the form $n^2+1$. My question is: does $$\lim_{k \to \infty} s_k/p_k^2$$ Exist, and if so, what does it converge to?
I conjecture that it does, though I had an unusual amount of trouble finding information on the asymptotics for primes of the form $n^2+1$. I suspect a work of Bateman and Horn to be useful here, but it's a little high-level for me.
I computed the first ~two million or so primes of the form $n^2+1$ and plotted $s_k/p_k^2$, for your leisure.
There is a conjectured asymptotic formula (indeed from the Bateman–Horn conjecture) for the number of positive integers $k\le x$ for which $k^2+1$ is prime, of the form $\sim Cx/(\log x)$ for a particular positive constant $C$. This implies that the $n$th such integer $k$ has size asymptotically equal to $\sim \frac1C n(\log n)$, and thus that the $n$th such prime $k^2+1$ has size asymptotically equal to $\sim \frac1{C^2} n^2(\log n)^2$. Since $p_n \sim n\log n$ by the prime number theorem, we see that $s_n/p_n^2$ conjecturally tends to the constant $\frac1{C^2}$.
I can't recover it now, but the constant $C$ can be estimated explicitly, so that should give a numerical value for the constant $\frac1{C^2}$ that could be checked against your numerical explorations.