Progress towards a nonintegrality conjecture (Q2663795)

Define the function \(S_r: \mathbb{N} \to \mathbb{Q}\) by \[ S_r(n) = \sum_{k=0}^n \frac k{k+r} \binom nk, \] for nonnegative integers \(r\). The authors first prove that \(S_r(n)\) is never an integer for any \(n\) when \(1 \leq r \leq 22\). Their main result is that the upper asymptotic density of integers \(n\) such that \(S_r(n)\) is an integer decays faster than any power of \(1/r\) as \(r\) grows.