Given any $k \in \mathbb{Z}$, does there exist solutions to $\sigma(n)=kn$, where $\sigma(n)$ is the sum of divisors of $n$?
Perfect numbers correspond to $k=2$. Is there some sort characterization of these numbers?
$\text{For instance, similar to the following statement:}$
$\text{$n$ is an even perfect number iff $n=2^{p-1}(2^p-1)$, where $2^p-1$ is a prime.}$
$\text{For $k=3$, for $n \leq 100,000$, $n=120$ and $n=672$ satisfy.}$
$\text{For $k=4$, for $n \leq 100,000$, $n=30240$ and $n=32760$ satisfy.}$
Interestingly for both these case, we have $k \mid n$ (similar to the conjecture that all perfect numbers are even).
$\text{The }$asymptotic$\text{ for the sum of divisors function does not rule out $k$ has to be finite.}$
As far as I know, existence of $k$-perfect numbers is open except for specific small $k$.
Your second conjecture, though, is false: $n=459818240$ satisfies $\sigma(n)=3n$ but $n\equiv 2\pmod 3$.