A certain reciprocal power sum is never an integer

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 $nge 2$. Now let deg$f(x)ge 2$. Clearly, $0<H_f(mathcal{S}_n)<zeta(2)<2$. But it is not clear whether the reciprocal power sum $H_f(mathcal{S}_n)$ can take 1 as its value. In this paper, with the help of a result of ErdH{o}s, we use the analytic and $p$-adic method to show that for any infinite sequence $mathcal{S}$ of positive integers and any positive integer $nge 2$, $H_f(mathcal{S}_n)$ is never equal to 1. Furthermore, we use a result of Kakeya to show that if $frac{1}{f(k)}lesum_{i=1}^inftyfrac{1}{f(k+i)}$ holds for all positive integers $k$, then the union set $bigcuplimits_{mathcal{S}in (mathbb{Z}^+)^{infty}} { H_f(mathcal{S}_n) | nin mathbb{Z}^+ }$ is dense in the interval $(0,alpha_f)$ with $alpha_f:=sum_{k=1}^{infty}frac{1}{f(k)}$. It is well known that $alpha_f= frac{1}{2}big(pi frac{e^{2pi}+1}{e^{2pi}-1}-1big)approx 1.076674$ when $f(x)=x^2+1$. Our dense result infers that when $f(x)=x^2+1$, for any sufficiently small $varepsilon >0$, 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 $1<H_f(mathcal{S}^{(2)}_{n_2})<1+varepsilon$.