Is there more than one harmonic number that is natural?

Question

n-th harmonic number is:

$H_n=\sum_{k=1}^n\frac1k$

is there some $n\neq1$ for which $H_n$ is a natural number?

Or can it be proven that there is no such number?

Answer 1

There are no examples other than $n=1$. This can be seen as a result of Bertrand's Postulate. Moreover, for all $n \geq 2$ the numerator of $H_{n}$ is an odd number while the denominator of $H_{n}$ is an even number.

Here is a fairly elementary proof.