Progress towards a nonintegrality conjecture

Abstract

Given $r \in \mathbb {N}$, define the function $S_{r}:\mathbb {N} \rightarrow \mathbb {Q}$ by

In 2015, the second author conjectured that there are infinitely many $r \in \mathbb {N}$ such that $S_{r}(n)$ is nonintegral for all $n \geqslant 1$, and proved that $S_{r}(n)$ is not an integer for $r \in \{2,3,4\}$ and for all $n \geqslant 1$. In 2016, Florian Luca and the second author raised a stronger conjecture that for any $r \geqslant 1$, $S_{r}(n)$ is nonintegral for all $n \geqslant 1$. They proved that $S_{r}(n)$ is nonintegral for $r \in \{5,6\}$ and that $S_{r}(n)$ is not an integer for any $r \geqslant 2$ and $1 \leqslant n \leqslant r-1$. In particular, for all $r \geqslant 2$, $S_{r}(n)$ 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 $n \geqslant 1$, and derived an algorithm that sometimes determines such nonintegrality; along the way he proved that $S_{r}(n)$ is nonintegral for $r \in \{7,8,9,10\}$ and for all $n \geqslant 1$. By improving this algorithm we prove the conjecture for $r\leqslant 22$. 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.