  Given $r \in \mathbb{N}$, define the function $S_{r}: \mathbb{N} \rightarrow
\mathbb{Q}$ by $S_{r}(n)=\displaystyle \sum_{k=0}^{n} \frac{k}{k+r}
\binom{n}{k}$. In $2015$, the second author conjectured that there are
infinitely many $r \in \mathbb{N}$ such that $S_{r}(n)$ is nonintegral for all
$n \geq 1$, and proved that $S_{r}(n)$ is not an integer for $r \in \{2,3,4\}$
and for all $n \geq 1$. In $2016$, Florian Luca and the second author raised
the stronger conjecture that for any $r \geq 1$, $S_{r}(n)$ is nonintegral for
all $n \geq 1$. They proved that $S_{r}(n)$ is nonintegral for $r \in \{5,6\}$
and that $S_{r}(n)$ is not an integer for any $r \geq 2$ and $1 \leq n \leq
r-1$. In particular, for all $r \geq 2$, $S_{r}(n)$ 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 $n \geq 1$, 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 $n \geq 1$.
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.
