Question 171
Denote by $\sigma(n)$ the sum of divisors of $n$. A positive integer $n$ ($n \ge 2$) is redundant if for any integer $k$, with $k \lt n$, we have $\frac{\sigma(k)}{k} \lt \frac{\sigma(n)}{n}$. (For instance if $n = 1, 2, 3, 4, 5$, then $\frac{\sigma(n)}{n} = 1, \frac{3}{2}, \frac{4}{3}, \frac{7}{4}, \frac{6}{5}$ respectively. Hence, $2$ and $4$ are redundant, $3$ and $5$ are not.) Show that there exist infinitely many redundant numbers.