Here is a problem concerning (hopefully!) Harmonic numbers:
Suppose that someone 'partitions' a natural number $n>0$ into $j$ 'parts' $m_k$, such that $\sum\limits_{k=1}^j m_k=n$. Since harmonic numbers series diverge, I conclude that also $\sum_{k=1}^j \frac{1}{m_k}$ diverges. The question is "is there a way to relate the last sum with the harmonic numbers?"