WE MUST BE PERFECT!

Call the perfect number $x$

Factors of $x$ are $F_1 , F_2 , F_3 … F_n$

$\sum_{i=1}^n F_n = 2x$

$\frac{1}{F_1} + \frac{1}{F_2} … + \frac{1}{F_n} = \frac{\frac{x}{F_1} + \frac{x}{F_2} … + \frac{x}{F_n}}{x}$

Notice that because $x$ is a perfect number, $\frac{x}{F_1} + \frac{x}{F_2} … + \frac{x}{F_n}$ is equal to $\sum_{i=1}^n F_n = 2x$

$\frac{2x}{x} = \boxed{2}$

It's true since,

$\frac{n + \dots + c + b + a}{n} = 2n \implies \frac{n}{a} + \frac{n}{b} + \dots = 2n \implies \frac{1}{a} + \frac{1}{b} + \dots = 2$