(Sorry for my poor english..)

I have a question in number theory. (Just my curiosity)

Let $f(z)\in \mathbb{Z}[x]$ be an irreducible polynomial with degree is larger than or equal to 2. Then is the number \begin{equation} \sum_{n=1}^{\infty} \frac{1}{f(n)} \end{equation} irrational? Or there is an irreducible polynomial $f(x)\in \mathbb{Z}[x]$ such that this number is rational?

- More generally, if for all $n\in \mathbb{N}$, $f(x)$ and $f(x+n)$ are relativlely prime in $\mathbb{Z}[x]$, then is it true..?

I also know that proving the irrationality is very difficult, for example $\zeta(2k+1)$...

I could not find references about this problem.. Has this problem been proved? Or, is there a name for this problem?