A new function associated with the prime factors of $(^{n}_{k})$

Abstract:

Let $g(k)$ denote the least integer $> k + 1$ so that all the prime factors of $\left ( {\begin {array}{*{20}{c}} {g(k)} \\ k \\ \end {array} } \right )$ are greater than k. The irregular behavior of $g(k)$ is studied, obtaining the following bounds: ${k^{1 + c}} < g(k) < \exp (k(1 + o(1))).$ Numerical values obtained for $g(k)$ with $k \leqq 52$ are listed.