If an odd perfect number exist could be a solitary number?

Question

Perfect numbers is a number that is half the sum of all of its positive divisors .And solitary numbers means that $\frac {\sigma(n)}{n}$ is an irreducible fraction, it's seems to me that all even perfect numbers are irrelevent from the list of solitary numbers, then i come up with this question: What about odd perfect number if it exist then could be a solitary number ?

Answer 1

If $n$ is a perfect number then $n| \sigma(n)$ .

A number is solitary when $n$ and $\sigma(n)$ are coprime.

The only way in which a number is perfect and solitary at the same time is if $n| gcd(n, \sigma(n))$ and $gcd(n, \sigma(n))=1$, meaning $n=1$.

But $n=1$ is not perfect.