Progress towards a nonintegrality conjecture

Abstract

Given $r\in \mathbb{N}$, define the function $S_r:\mathbb{N}\to \mathbb{Q}$ by $S_r(n)=\sum _{k=0}^n\frac{k}{k+r}\left(\genfrac{}{}{0ex}{}{n}{k}\right)$. In $2015$, the second author conjectured that there are infinitely many $r\in \mathbb{N}$ such that $S_r(n)$ is nonintegral for all $n\ge 1$, and proved that $S_r(n)$ is not an integer for $r\in \{2,3,4\}$ and for all $n\ge 1$. In $2016$, Florian Luca and the second author raised the stronger conjecture that for any $r\ge 1$, $S_r(n)$ is nonintegral for all $n\ge 1$. They proved that $S_r(n)$ is nonintegral for $r\in \{5,6\}$ and that $S_r(n)$ is not an integer for any $r\ge 2$ and $1\le n\le r-1$. In particular, for all $r\ge 2$, $S_r(n)$ is nonintegral for at least $r-1$ values of $n$. In $2018$, the fourth author gave sufficient conditions for the nonintegrality of $S_r(n)$ for all $n\ge 1$, and derived an algorithm to sometimes determine such nonintegrality; along the way he proved that $S_r(n)$ is nonintegral for $r\in \{7,8,9,10\}$ and for all $n\ge 1$. By improving this algorithm we prove the conjecture for $r\le 22$. Our principal result is that $S_r(n)$ is usually nonintegral in that the upper asymptotic density of the set of integers $n$ with $S_r(n)$ integral decays faster than any fixed power of $r^{-1}$ as $r$ grows.