Problem 1063

Erdős Problem #1063: declared status 'open'. Formalized: yes. Let $k\geq 2$ and define $n_k\geq 2k$ to be the least value of $n$ such that $n-i$ divides $\binom{n}{k}$ for all but one $0\leq i<k$. Estimate $n_k$. Current best: Erd\H{o}s and Selfridge noted (and a proof can be found in [Mo85]) that if $n\geq 2k$ then there must exist at least one $0\leq i<k$ such that $n-i$ does not divide $\binom{n}{k}$. Monier [Mo85] observed that $n_k\leq k!$ for $k\geq 3$, since $\binom{k!}{k}$ is divisible by $k!-i$ for $1\leq i<k$. Cambie observes in the comments that this can be improved to$$n_k\leq k[2,3,\ldots,k-1]\leq e^{(1+o(1))k},$$where $[\cdots]$ is the least common multiple. Prize: no. OEIS: A389360. Tags: number theory.