A certain reciprocal power sum is never an integer

Abstract

By $(\mathbb{Z}^+)^{\infty}$ we denote the set of all the infinite sequences $\mathcal{S}=\{s_i\}_{i=1}^{\infty}$ of positive integers (note that all the $s_i$ are not necessarily distinct and not necessarily monotonic). Let $f(x)$ be a polynomial of nonnegative integer coefficients. Let $\mathcal{S}_n:=\{s_1, ..., s_n\}$ and $H_f(\mathcal{S}_n):=\sum_{k=1}^{n}\frac{1}{f(k)^{s_{k}}}$. When $f(x)$ is linear, Feng, Hong, Jiang and Yin proved in [A generalization of a theorem of Nagell, Acta Math. Hungari, in press] that for any infinite sequence $\mathcal{S}$ of positive integers, $H_f(\mathcal{S}_n)$ is never an integer if $n\ge 2$. Now let deg$f(x)\ge 2$. Clearly, $0 0$, there are positive integers $n_1$ and $n_2$ and infinite sequences $\mathcal{S}^{(1)}$ and $\mathcal{S}^{(2)}$ of positive integers such that $1-\varepsilon<H_f(\mathcal{S}^{(1)}_{n_1})<1$ and $1<H_f(\mathcal{S}^{(2)}_{n_2})<1+\varepsilon$.