# Tagged Questions

A perfect number is a positive integer that is equal to the sum of its proper positive divisors.

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### If $N = q^k n^2$ is an odd perfect number, if $n < q^{k+1}$, does it follow that $k > 1$?

Let $N = q^k n^2$ be an odd perfect number with Euler prime $q$. According to Dickson (as pointed out recently by Beasley), Descartes conjectured $k=1$ in a letter to Mersenne in 1638, with Frenicle'...
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### Is $\liminf \frac{\sigma_{k}(({2}^{m-1})({2^m-1}))}{\phi_{k}(({2}^{m-1})({2^m-1}))}$ finite for every $k$?

I would like to check if this limit :$$\liminf \frac{\sigma_{k}(({2}^{m-1})({2^m-1}))}{\phi_{k}(({2}^{m-1})({2^m-1}))}$$ finite for every $k$? where :$\phi_{k}$ is iterating Euler - totient function ...
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### If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, is $n$ a square?

(I have asked a similar question in MSE four days ago, but did not receive any answers. I have therefore cross-posted it to this site, hoping to get some responses.) An odd perfect number $N$ is ...
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### If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, is $n$ squarefree?

(I have asked a similar question in MSE around a week ago, but did not receive any responses. I have therefore cross-posted it to this site, hoping to get some answers.) An odd perfect number $N$ is ...
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### Diagonal argument for even perfect numbers

Following this, let's define the notion of perfect sequence as follows: $(u_{i})_{i}$ is a perfect sequence if and only if it is the sequence of divisors of an even perfect number in increasing order ...
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### On odd perfect numbers $N = q{p^{2a}}{m^2}$ satisfying certain conditions

Let $N = q{p^{2a}}{m^2}$ be an odd perfect number, satisfying the conditions $$\sigma(m^2) = p^{2a}$$ $$\sigma(p^{2a}) = q$$ and $$q + 1 = 2{m^2}.$$ Note the following: $p^a m < q$ $q$ is ...
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### If $q^k n^2$ is an odd perfect number with Euler prime $q$, are the following statements known to hold in general? [closed]

(Note: This has been cross-posted to MSE. However, I feel that it is more likely to receive a good answer here, because I believe that it is a research-level question. For the mathematicians who ...
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### What are the divisors of $2n^2 - \sigma_{1}(n^2)$ for composite $n$?

What are the divisors of $2n^2 - \sigma_{1}(n^2)$ for composite $n$? Here, $\sigma_{1}$ is the classical sum-of-divisors function. For example, $\sigma_{1}(3^2) = 1 + 3 + {3^2} = 13$. (The function ...
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### What is the latest progress in the research on Odd Perfect numbers? [closed]

What is the latest progress in the research on Odd Perfect numbers? I may be wrong, but I found a little on Perfect numbers in the latest issues of SCI journals. Is it really so? I would like to have ...
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### Generalized quasi-perfect numbers

A number $n \in \mathbb{N}$ is called quasi-perfect if $\sigma(n) = 2n+1$, where $\sigma$ is the sum of divisors function. It is known that if $n$ is quasi-perfect, then it must be the square of an ...
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### Proof of the Infinitude of Odd Primitive Pseudoperfect Numbers

I'm interested in the infinitude of odd primitive pseudoperfect numbers. Richard K. Guy's book "Unsolved Problems in Number Theory 3rd edition" says that P. Erdős proved the infinitude of odd ...
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### Integers n such that sigma(n)=omega(n)n and omega(n) divides n

Are there other integers $n$ than even perfect numbers such that $\sigma(n)=\omega(n)n$ and $\omega(n)\vert n$? Thanks in advance.
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### On J. T. Condict's Senior Thesis on Odd Perfect Numbers

I am trying to locate a copy of J. T. Condict's senior thesis on odd perfect numbers: J. Condict, On an odd perfect number's largest prime divisor, Senior Thesis, Middlebury College (1978). I am ...
### Re: Mordell's Equation $y^2 = x^3 + k$ and Perfect Numbers
I have already tried a somewhat exhaustive search of the literature, but couldn't find anything close to the problem that I am working on. My question is: When does Mordell's Equation y^2 = x^3 +...
In the PhD dissertation titled "Algorithms in the Study of Multiperfect and Odd Perfect Numbers" (hyperlinked here) and completed in 2003, Ronald Sorli conjectured that the exponent $k$ on the Euler ...