323. Interesting numbers. If we sum the reciprocals of all divisors of the number 6, we get $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=2$. Find two other numbers for which the analogous sum is also equal to 2.
323. Let $1, a, b, \ldots, n$ be the divisors of the number $n$, listed in ascending order; suppose that the equality

$$
\frac{1}{1}+\frac{1}{a}+\frac{1}{b}+\cdots+\frac{1}{n}=2
$$

holds. Multiplying both sides of this equation by $n$, we get

$$
\frac{n}{a}+\frac{n}{b}+\cdots+1=n
$$

where the left side represents the sum of the proper divisors of the number $n$, listed in descending order. But this means that $n$ is a perfect number. The next perfect numbers after 6 are 28 and 496.

[D. Silverman, M. M., 44, 56 (January 1971).]