It is unknown if odd perfect numbers exist and it is known that the even perfect numbers are those of the form $$2^{n-1}(2^n-1)$$, where $2^n-1$ is a (Mersenne-)prime.
But what is known about the multi-perfect numbers ?
Is it known whether
for all $k\ge 3$, there is a number $n$ with $\frac{\sigma(n)}{n}=k$ ?
Examples are
$k=2$ : $6$
$k=3$ : $120$
$k=4$ : $30\ 240$
$k=5$ : $14\ 182\ 439\ 040$
$k=6$ : $154\ 345\ 556\ 085\ 770\ 649\ 600$
there are infinite many numbers $n$ such that $\frac{\sigma(n)}{n}$ is an integer $k\ge 3$ ? For $k=2$, this is closely related to the Mersenne-prime-conjecture that there are infinite many mersenne primes and therefore infinite many perfect numbers.
How can very large multi-perfect numbers (excluding perfect numbers) be constructed ?
Have a look at The Multiply Perfect Numbers Page, maintained by Achim Flammenkamp, although the date specified at the very bottom of the home page is Jan. 25, 2014 (which I presume, denotes the last time that the page was modified).
You can also check out Ron Sorli's thesis here, for a discussion of some algorithms on searching for multiperfect numbers.