On a nonintegrality conjecture

Abstract

It is conjectured that the sum $$S_r(n)=\sum _{k=1}^n\frac{k}{k+r}\left(\genfrac{}{}{0ex}{}{n}{k}\right)$$ for positive integers $r,n$ is never integral. This has been shown for $r\le 22$. In this note we study the problem in the ``$n$ aspect" showing that the set of $n$ such that $S_r(n)\in \mathbb{Z}$ for some $r\ge 1$ has asymptotic density $0$. Our principal tools are some deep results on the distribution of primes in short intervals.