As tell us the Wikipedia section dedicated to *Odd perfect numbers* (please, see also the related references if you need it), any perfect number has the form $$n=q^\alpha m^2$$
where the integer $\alpha\geq 1$ satisfies $\alpha\equiv 1\text{ mod }4$, the integer $q$ is a prime number satisfying $q\equiv 1\text{ mod }4$ and the positive integer $m$ satisfies $\gcd(q,m)=1$. This is the Euler's theorem for odd perfect numbers.

The factor $q^\alpha$, in the factorization of Euler's theorem for odd perfect numbers, is the so-called Euler factor of our odd perfect number $n$.

In this paragraph we remember the definitions/notations for some number theoretic function: for real numbers $x\geq 1$, in this post the prime-counting function is denoted as $\pi(x)$. For an integer $n>1$, we denote its greatest prime factor as $\operatorname{gpf}(n)$, and the product of the distinct prime numbers dividing $n$ as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$
with the definition $\operatorname{rad}(1)=1$ (see it you want the Wikipedia *Radical of an integer* and the corresponding article from MathWorld *Greatest Prime Factor*).

Question.Is it possible, under the assumption that $n$ is an odd perfect number, to get bounds for the inequality $$\text{lower bound}<\pi\left(q^\alpha\right)<\text{upper bound},\tag{1}$$ being $q^{\alpha}$ its Euler factor, or well for inequalities of the type $$\text{lower bound}<\pi\left(\operatorname{rad}(n)\right)<\text{upper bound},\tag{2}$$ or $$\text{lower bound}<\pi\left(\operatorname{gpf}(n)\right)<\text{upper bound}\,?\tag{3}$$Many thanks.

The calculations that I evoke are deductions of the bounds as functions of $n$ (including constants, if it is the case). My motivation is to learn how do it, and that if there is some answer it can be a very good reference of your proposition/s for other people interested in this theory, I think.

such bounds also can be written in terms of the factorization of our odd perfect numbers, I am saying that you can express your bounds invonving the primes of the primes and their corresponding exponents of the prime factorization of $n$. Many thanks again y thanks for the patience of all users. $\endgroup$