Background
Let $\sigma(x)$ be the sum of divisors of the positive integer $x$. A number $l$ is called perfect if $\sigma(l)=2l$.
Let $n$ be an odd perfect number given in the so-called Eulerian form $n = p^k m^2$ where $p$ is the special/Euler prime satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$.
Motivation
Has anybody seen
Enrique Santos L's "Proof that no odd perfect number exists"?
From Eq. (6) in that paper, we have $$p^x a = \sigma(a) \frac{\sigma(p^x)}{2},$$ where it is implied that $a = m^2$ and $x = k$ (to use our notation).
Then in the section Separation in two equations in that paper, Enrique claims that $\sigma(a)$ has to be coprime to $a$, which I know to be false since $$\gcd(m^2,\sigma(m^2))=\frac{\sigma(m^2)}{p^k}=\frac{m^2}{\sigma(p^k)/2} \geq 3,$$ a result of Dris from 2012.
Inquiry
Can the rest of the "proof argument" be salvaged? Is it possible to mend Enrique's "proof argument" to hopefully produce some partial results on odd perfect numbers?
Eulerian form is not used in the paper "Proof that no odd perfect number exists", as it is not used the assumption that $n$ has to be odd. Then it is not correct the claim that "it is implied that $a = m^2$", which would be valid only under the odd $n$ assumption. That is why your result cannot be applied also. In fact, $σ(a)$ is co-prime to $a$ in every known perfect number.
The cited paper has an important flaw, anyway, that is the affirmation that "it should be obvious" that the following relation
$$ \begin{align} \ p^x = \prod_i {σ(q_i^{s_i + r_i}) \over q_i^{r_i}} = {\prod_i ( 1 + q_i + ... + q_i^{s_i + r_i} ) \over \prod_i q_i^{r_i}} \end{align} $$
can not be integer "unless the denominator is 1". It is not obvious at all, and it has not been proved later also.