Is there a set $A$ of positive integers such that

  - $\sum_{n \in A} \frac{1}{n} = \infty$, and

  - there is no polynomial $f \in \mathbb{Z}[x]$ of degree at least $2$
    which takes infinitely many values in $A$?

<b>Added on Feb 16, 2015:</b> Seva gave a complete answer to this question;
though a little later, he claimed a <b>great deal</b> more -- his claim amounts
to the following: 

> <b>Claim:</b> There is a partition of $\mathbb{N}$ into
>
>   - a set $A$ of asymptotic density $1$ of *'non-values of non-linear polynomials'*,
>     which has finite intersection with the image of any polynomial $f \in \mathbb{Z}[x]$
>     of degree $\geq 2$, and
>
>   - a set $B$ of asymptotic density $0$ of *'values of non-linear polynomials'*,
>     which contains all but finitely many positive values taken by any polynomial
>     $f \in \mathbb{Z}[x]$ of degree $\geq 2$.

If Seva is right, can the construction of such partition of $\mathbb{N}$
be made more explicit?