Question 113
Let $p \gt 200$ be a prime number. We call a positive integer $n$ *good* if $p$ divides the numerator of the irreducible fraction $\frac{a_n}{b_n} = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$. Prove that for all large enough $N$ the number of good numbers not exceeding $N$ is not greater than $C N^{\frac{3}{4}}$, where $C$ is a constant; possibly depends on $p$.