Inelegant Explorations of Some Personal Problems

Harmonic Integers

Years back, I encountered the following problem in a book on Number Theory: Prove that the sum $H_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$ is never an integer after $n > 1$. In other words, in the sequence $1, 1+\frac{1}{2}, 1+\frac{1}{2}+\frac{1}{3}, \dots$, there is only one integer, and that is the first number $1$.