On the integrality of the elementary symmetric functions of $1,1\mathrm{/}3,\mathrm{.}\mathrm{.}\mathrm{.},1\mathrm{/}(2n-1)$

Erdos and Niven proved that for any positive integers $m$ and $d$, there are only finitely many positive integers $n$ for which one or more of the elementary symmetric functions of $1\mathrm{/}m,1\mathrm{/}(m+d),\mathrm{.}\mathrm{.}\mathrm{.},1\mathrm{/}(m+nd)$ are integers. Recently, Chen and Tang proved that if $n\ge 4$, then none of the elementary symmetric functions of $1,1\mathrm{/}2,\mathrm{.}\mathrm{.}\mathrm{.},1\mathrm{/}n$ is an integer. In this paper, we show that if $n\ge 2$, then none of the elementary symmetric functions of $1, 1/3, …, 1/(2n-1)$ is an integer.