Timeline for Recovering n from sigma(n)/n
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 15, 2011 at 3:47 | answer | added | Michel Marcus | timeline score: 1 | |
| Feb 24, 2011 at 8:42 | vote | accept | Tom De Medts | ||
| Feb 23, 2011 at 11:49 | answer | added | Gjergji Zaimi | timeline score: 14 | |
| Feb 23, 2011 at 10:42 | comment | added | Tom De Medts | @Francois: Something more general is true indeed; namely if $m \mid n$, then $\delta(n) \geq \delta(m)$. Your observation is the case $m=6$. | |
| Feb 23, 2011 at 10:21 | comment | added | François Brunault | We have $\frac{\sigma(n)}{n} \geq \prod_p (1+\frac{1}{p})$ where $p$ runs over the prime factors of $n$. In particular if $n$ is divisible by $6$ then $\sigma(n)/n \geq 2$ ($n$ is abundant). This shows that some values like $\frac{7}{6}$ or $\frac{11}{6}$ are never attained. | |
| Feb 23, 2011 at 10:18 | comment | added | Tom De Medts | @Charles: good point; this part of the question is indeed too ambitious. | |
| Feb 23, 2011 at 10:11 | comment | added | François Brunault | @Charles : the question of finding all $n$ such that $\delta(n)=q$ seems indeed very difficult, but at least one can try to prove some qualitative results on $\delta$. For example, is there an integer $N$ such that $\{n \in \mathbb{N} | \delta(n) \in \frac{1}{N} \mathbb{Z}\}$ is infinite ? | |
| Feb 23, 2011 at 10:02 | comment | added | Charles Matthews | The case of the value 2 being perfect numbers, the question seems over-ambitious. | |
| Feb 23, 2011 at 9:56 | history | asked | Tom De Medts | CC BY-SA 2.5 |