Let $\{r_i\}_{i \in \mathbb{N}}$ be a periodic sequence of integers. That is, there exists a $t$, such that, for all $i \in \mathbb{N}$, $r_i = r_{i+t}$. What I need is a large prime dividing the numerator of $\displaystyle \sum_{i=1}^n \dfrac{r_i}{i}$. I general, I'd like to know more about the largest prime dividing that sum as $n$ goes to infinity. In particular, I would love to know a proof (or even more spectacular, a counterexample) to the following conjecture:

For every $c \in \mathbb{R}$, there exists a positive integer $n \ge 2$ and a prime $p > cn$ that divides the numerator of $\displaystyle \sum_{i=1}^n \dfrac{r_i}{i}$

Furthermore, I metaconjecture that my effort over the last 6 months to prove this, will be nullified within an hour or two.

If at least one of my conjectures turns out to be false, for which $c > 0$ is my first conjecture true and easily provable?