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I know that Descartes is considered to be the first to ask whether or not odd perfect numbers exist ($n$ such that $\sigma(n)=2n$, where $\sigma(n)$ is the sum of divisors of $n$), and he also discovered several multiply-perfect numbers ($n$ such that $\sigma(n)=kn$ for some integer $k\geq 2$). No odd multiply-perfect numbers greater than one are yet known, and I've heard it called a conjecture that they don't exist several times, but I haven't found an attribution to anyone in particular. Is there an agreed upon origin for the conjecture?

The context of my question (please, feel free to skip this):

I'm looking for someone to attribute in a paper I'm writing that makes a refinement of this form of conjecture for a generalization of the multiply-perfect numbers. Specifically, let $$\dfrac{\sigma(n)}n=\dfrac{k}m$$ where $\gcd(k,m)=1$ and $m$ is a practical number i.e. $m=1$, or $p_i^{a_i+1}\leq 1+\sigma(p_1^{a_1}p_2^{a_2}...p_i^{a_i})$ for every $i\in[1,\omega(m)]$, where $p_1^{a_1}p_2^{a_2}...p_{\omega(m)}^{a_{\omega(m)}}$ is the canonical prime factorization of $m$. I call these pad (practical abundancy denominator) numbers, and conjecture that every such number is practical (i.e. a practical number). $\dfrac{\sigma(n)}n$ is called the abundancy of $n$.

It is easy to see that every even perfect number is practical, and every practical number greater than $1$ is trivially even, so the conjecture that every perfect number is practical is equivalent to the conjecture that they're all even. From this perspective it's natural to ask whether every multiply-perfect number is practical. I've tested essentially every known multiply-perfect number using the factorizations from Achim Flammenkamp's database and all of them so far are practical, so already this seems like a promising lead.

Clearly the multiply-perfect numbers are the special case of the displayed equation where $m=1$, which is trivially practical, so every multiply-perfect number is a pad number. For an idea of their relative size, there are $6484$ pad numbers less than $10^6$, all of them practical, compared with only $10$ multiply-perfect numbers in the same range. An additional fact that makes this conjecture seem quite intuitive is that all small multiples of a practical number are also practical. Specifically, if $a$ is practical and $b\leq \sigma(a)+1$ then $ab$ is practical. Since, $k/m$ is just the abundancy of $n$ in lowest terms, there exists some positive integer $r$ such that $n/m=\sigma(n)/k=r$.

When $r=n$, the multiply perfect numbers, we have no idea why every term would be practical, and yet this is the special case I've tested the furthest, and no counterexamples have yet been found. On the other hand, if $r\leq 1+\sigma(m)$ then $n$ is obviously practical. While in retrospective it seems rather intuitive to me, this was a purely numerical observation. Please let me know if you have ideas about this problem, since I believe it has the potential to remain open for a long time, but someone to attribute is what I'm asking for, since right now I feel I have very few people to attribute in my paper. This is for my first paper, which I think I will try to have published in Experimental Mathematics, since computational experiments were the method by which I reached this generalization. If you have strong opinions about other papers or people who should be attributed then please mention them at least in the comments. I'm not looking to downplay anyone's contributions, and I realize the importance of showing how your work ties in with that of others.

Edit: Note that the same conjecture appears to hold with $\sigma(m)$ replaced by $\sigma^*(m)=\prod_{i=1}^{\omega(m)}(p_i^{a_i}+1)$, the sum of the unitary divisors of $n$, which might be simpler to work with.

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The earliest reference seems to be from 1966: E.A. Bugulov, On the question of the existence of odd multiperfect numbers, Kabardino-Balkarskaya State University Učen. Zap. 30 (1966) 9-19. [I could not find this article online.]

Bugulov showed that an odd multiperfect number must have at least 11 distinct prime divisors, more precisely, odd $k$-perfect numbers contain at least $\omega$ distinct prime factors, where $(k,\omega) =$ (3, 11), (4,21), (5, 54),... This result is discussed and improved by Shigeru Nakamura, On k-perfect numbers, Journal of the Tokyo University of Mercantile Marine (Natural Sciences), 33 (1982) 43–50. [Listed here, but not available online.]

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  • $\begingroup$ Thank you very much. Are you pretty confident that this was the first reference, of should I keep the question open a little longer in case someone thinks they are aware of an earlier one? Does Bugulov claim the generalized conjecture as his own? I want to make sure I present an accurate history of the problem. $\endgroup$ – Jaycob Coleman Nov 28 '13 at 22:21
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    $\begingroup$ my source is page 49 of "Unsolved problems in number theory", online here: librarum.org/book/882/65 . I have no earlier reference, but there may well be one, and since I have no access to this obscure paper of Bugulov, I cannot say much more. It seems clear enough that Bugulov spent quite some effort to show that odd multiperfect numbers are rare . That would seem enough to credit him with this conjecture, in the absence of an older source. $\endgroup$ – Carlo Beenakker Nov 29 '13 at 9:13

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