Show that for $n\gt 2$, $\frac{\sigma_1(n)}{n}\lt H_n$

Question

Is there any positive integer $n$, besides $n=2$ such that $$\frac{\sigma_1(n)}{n}=H_n$$

They are clearly asymptotic from their graphs so can we show that for $n\gt 2$, $$\frac{\sigma_1(n)}{n}\lt H_n$$

Answer 1

For $n \ge3$, $$\frac{\sigma_1 (n)}{n} = \sum_{d|n}\frac{d}{n} = \sum_{d|n}\frac{1}{d} < \sum_{d=1}^n \frac{1}{d}=H_n$$