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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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  • $\begingroup$ The references here look like a good place to start: oeis.org/A007691 $\endgroup$ Commented May 11, 2015 at 22:09
  • $\begingroup$ @Peter, I am posting a follow-up answer in a bit. $\endgroup$ Commented Feb 3, 2020 at 8:19
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    $\begingroup$ From OEIS sequence A007691, T. D. Noe notes that: "Luca's solution of problem 11090, which proves that for $k>1$ there are an infinite number of $n$ such that $n$ divides $\sigma_k(n)$, does not apply to this sequence. However, it is conjectured that this sequence is also infinite. - T. D. Noe, Nov 04 2007". $\endgroup$ Commented Feb 4, 2020 at 8:33

2 Answers 2

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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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This is a partial answer, in response to your second question.

The following appears as Exercise #9 in page 226 of the Seventh Edition of David M. Burton's book, Elementary Number Theory:

If $\sigma(n)=kn$, where $k \geq 3$, then the positive integer $n$ is called a $k$-perfect number (sometimes, multiply perfect).

For each $k$, it is conjectured that there are only finitely many $k$-perfect numbers. The largest one discovered has $558$ digits and is $9$-perfect.

The Seventh Edition of Burton's Elementary Number Theory is copyrighted year $2011$.

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  • 1
    $\begingroup$ From OEIS sequence A007691, T. D. Noe notes that: "Luca's solution of problem 11090, which proves that for $k>1$ there are an infinite number of $n$ such that $n$ divides $\sigma_k(n)$, does not apply to this sequence. However, it is conjectured that this sequence is also infinite. - T. D. Noe, Nov 04 2007". $\endgroup$ Commented Feb 4, 2020 at 8:34

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