IR-TST 2007 Problem 7

Does there exist a a sequence $a_{0},a_{1},a_{2},\dots$ in $\mathbb N$ , such that for each $i\neq j, (a_{i},a_{j})=1$ , and for each $n$ , the polynomial $\sum_{i=0}^{n}a_{i}x^{i}$ is irreducible in $\mathbb Z[x]$ ? By Omid Hatami