About the sequence $d_n$ of the denominators of harmonic numbers, I know these facts:
It is unbounded, since $p\mid d_p$ for any prime $p$.
It contains only one $1$.
What more is known? Specially, is it true (as intuition suggests) that $$\lim_{n\to\infty}d_n=\infty$$ ? If not, is it known the inferior limit?
The denominator $d_n$ is divisible by every prime $p$ satisfying $p\leq n<2p$ (only one term in the sum $H_n$ has denominator divisible by $p$). Bertrand's Postulate implies such a prime exists if $n \geq 2$, so $d_n\geq n/2 \to\infty$ as $n\to\infty$.