Squarefree discriminants of polynomials with prime coefficients

In this paper, we consider the family of monic polynomials with prime coefficients and the family of all polynomials with prime coefficients. We determine the number of $f(x)$ in each of these families having: squarefree discriminant; $\mathbb{Z}[x]\mathrm{/}(f(x))$ as the maximal order in $\mathbb{Q}[x]\mathrm{/}(f(x))$.