# Questions tagged [perfect-numbers]

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

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### Improving the lower bound $I(n^2) > \frac{2(q-1)}{q}$ when $q^k n^2$ is an odd perfect number

Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Define the abundancy index $$I(x)=\frac{\sigma(x)}{x}$$ where $\sigma(x)$ ...
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### On odd perfect numbers and a GCD - Part II

(Note: A detailed version of this question was posted in MSE last April 15, 2020. It has not received any responses there as of yet. I have therefore cross-posted it here, hoping that it is ...
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### On the nearest-square function and the quantity $m^2 - p^k$ where $p^k m^2$ is an odd perfect number

This question has been cross-posted from this MSE question and is an offshoot of this other MSE question. (Note that MSE user mathlove has posted an answer in MSE, which I could not completely ...
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### Can an even perfect number be a sum of two cubes?

A similar question was asked before in https://math.stackexchange.com/questions/2727090/even-perfect-number-that-is-also-a-sum-of-two-cubes, but no conclusions were drawn. On the Wikipedia article of ...
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### If $q^k n^2$ is an odd perfect number with Euler prime $q$, is $\sigma(q^k)/n + \sigma(n)/q^k$ bounded from above?

STATEMENT OF THE PROBLEM If $q^k n^2$ is an odd perfect number with Euler prime $q$, is $\sigma(q^k)/n + \sigma(n)/q^k$ bounded from above? MOTIVATION Let $\sigma=\sigma_{1}$ denote the classical ...
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### Looking for an appropriate reference(s) for two conjectures on odd perfect numbers

(I apologize in advance if this question is unsuitable for MO. If so, please let me know and I will migrate it to MSE.) Let $\sigma(M)$ be the sum of the divisors of the positive integer $M$. For ...
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### If $N = q^k n^2$ is an odd perfect number, and $n < q^{k+1}$, 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 ...
### 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 the ...