Let $d,m$ and $n$ be positive integers. In 1915, Theisinger proved that if $n\ge 2$, then the $n$-th harmonic sum $\sum_{k=1}^n\frac{1}{k}$ is not an integer. In 1946, Erd\H{o}s and Niven extended Theisinger's theorem by showing that there are only finitely many positive integers $n$ for which one or more of the elementary symmetric functions of $1/m,1/(m+d),...,1/(m+nd)$ are integers. In 2015, Wang and Hong proved that none of the elementary symmetric functions of $1,1/3,...,1/(2n-1)$ is an integer if $n\ge 2$. In this paper, we show that if $n\ge 2$, then for arbitrary $n$ positive integers $s_0, ..., s_{n-1}$ (not necessarily distinct and not necessarily monotonic),the following multiple reciprocal power sum $$\sum\limits_{0\le i
