On a nonintegrality conjecture (Q2147392)

In this interesting and well-written paper, the authors deal with the following conjecture: for any positive integers \(r\), the number \[S_r(n) := \sum_{k=0}^n \frac{k}{k+r} \binom{n}{ k}\] is never integral for all positive integers \(n\). In [Eur. J. Math. 6, No. 4, 1496--1504 (2020; Zbl 1465.05007)], the authors proved the conjecture for \(1 \leqslant r \leqslant 22\), and that the set of positive integers \(n\) such that \(S_r(n) \in \mathbb{Z}\) has upper density \(\ll r^{-k}\) for any \(k \in \mathbb{Z}_{\geqslant 1}\). A further step is taken in this work with the proof by the authors that the set \(\mathcal{S} := \left\lbrace n \in \mathbb{Z}_{\geqslant 1} : S_r(n) \in \mathbb{Z} \ \text{for some} \ r \in \mathbb{Z}_{\geqslant 1} \right\rbrace\) has zero density as a subset of the integers and every number of \(\mathcal{S}\) is greater than \(10^6\). The proof makes use in a crucial way of profound results on prime numbers in short intervals, in particular the one given in [\textit{R. C. Baker} et al., Proc. Lond. Math. Soc. (3) 83, No. 3, 532--562 (2001; Zbl 1016.11037)]. As a consequence of this result, the authors deduce that the sum of reciprocal of members of \(\mathcal{S}\) converges. Note that the related alternating sum case is much simpler by virtue of the identity \[\sum_{k=0}^n (-1)^{k+1} \frac{k}{k+r} \binom{n}{k} = \frac{1}{\binom{n+r}{n}}.\]