The perfect-numbers tag has no wiki summary.

**2**

votes

**1**answer

91 views

### Solutions of $rad(\sigma(m))=2rad(m)$

For $m$ a positive integer greater than $1$, let $rad(m)$ be the product of all distinct primes dividing $m$. If $n$ is an odd perfect number (conjectured not to exist), one would have $\sigma(n)=2n$, ...

**3**

votes

**1**answer

159 views

### An equation involving perfect numbers

Let $s,x_1,x_2,\cdots, x_s$ be natural numbers not neccesarily distinct.
I am interested in solving the equation $$(x_1+x_2+\cdots +x_s)^s=2^s(x_1\cdot x_2\cdots x_s)^2$$
Some Notes:
I have found ...

**5**

votes

**1**answer

209 views

### Who is attributed with the conjecture that every multiply-perfect number greater than $1$ is even?

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 ...

**1**

vote

**1**answer

238 views

### Any results towards the irrationality of the sum of reciprocals of perfect numbers? [closed]

This question is a follow up to my comment to Sum of the reciprocal of perfect numbers. I would like to know which results have been published about the possible irrationality of the sum of ...

**1**

vote

**0**answers

110 views

### Determine whether if $n$ is a primitive pseudoperfect (semiperfect) number, then $\sigma(n)<2^{\sigma_0(n)}$

Determine whether if $n$ is a primitive pseudoperfect (semiperfect) number, then $\sigma(n)<2^{\sigma_0(n)}$.
$\sigma_k(n)$ is the division function and $\sigma(n)=\sigma_1(n)$. A number is ...

**0**

votes

**1**answer

145 views

### Are all known $k$-multiperfect numbers (for $k > 2$) not squarefree?

I asked the following question in MSE four ($4$) days ago, but so far nobody has posted an answer.
The gist of the question is as follows:
Are all known $k$-multiperfect numbers (for $k > ...

**1**

vote

**1**answer

349 views

### 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 ...

**1**

vote

**0**answers

148 views

### 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. ErdÅ‘s proved the infinitude of odd ...

**4**

votes

**3**answers

263 views

### 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.