# What is known about multi-perfect numbers?

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 ?