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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 ?

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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.

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