On the Elementary Symmetric Functions of $\{1,1/2,\dots ,1/n\}\backslash \{1/i\}$

Abstract

In 1946, P. Erdős and I. Niven proved that there are only finitely many positive integers $n$ for which one or more of the elementary symmetric functions of $1,1/2$, $\cdots ,1/n$ are integers. In 2012, Y. Chen and M. Tang proved that if $n⩾4$, then none of the elementary symmetric functions of $1,1/2,\cdots ,1/n$ are integers. In this paper, we prove that if $n⩾5$, then none of the elementary symmetric functions of $\{1,1/2,\cdots ,1/n\}\backslash \{1/i\}$ are integers except for $n=i=2$ and $n=i=4$.