I'm curious about if the following question is in the literature or what work can be done about it.
Denote the number of distinct primes dividing an odd perfect number $N$ with the arithmetic function $\omega(N)$. Wikipedia has this section dedicated to near-square primes from the article Landau's problems. Under the assumption that there exists an odd perfect number $N$, we denote the number of distinct near-squares primes dividing $N$ as $\omega_{\text{nsp}}(N)$, thus $\omega_{\text{nsp}}(N)\leq \omega(N)$.
Here I add as general reference that a near-square prime is a prime number of the form $p=n^2+1$, that's the sequence A002496 from The On-Line Encyclopedia of Integer Sequences.
Question. I would like to know if we can improve the previous lower bound and/or the upper bound, in this way $$\text{lower bound}<\omega_{\text{nsp}}(N)<\text{upper bound}\tag{1}$$ being these bounds functions of $N$, that's $\text{lower bound}=\text{lower bound}(N)$ and $\text{upper bound}$ denotes a function $\text{upper bound}(N)$. Many thanks.
I can to deduce obvious improvements for particular cases of Touchard's theorem, or when I consider Euler's theorem for odd perfect numbers. Also I know bounds for the radical of an odd perfect number, bounds for the Euler's totient function $\varphi(N)$, and I know the product formula representation for these arithmetic functions.