Problem 291

Erdős Problem #291: declared status 'open'. Formalized: no. Let $n\geq 1$ and define $L_n$ to be the least common multiple of $\{1,\ldots,n\}$ and $a_n$ by$$\sum_{1\leq k\leq n}\frac{1}{k}=\frac{a_n}{L_n}.$$Is it true that $(a_n,L_n)=1$ and $(a_n,L_n)>1$ both occur for infinitely many $n$? Current best: There is in fact a necessary and sufficient condition: a prime $p\leq n$ divides $(a_n,L_n)$ if and only if $p$ divides the numerator of $1+\cdots+\frac{1}{k}$, where $k$ is the leading digit of $n$ in base $p$. Prize: no. OEIS: A110566. Tags: number theory, unit fractions.