Does there exist a sequence of nonzero integers $a_n$ such that for all $n$ the polynomial $\sum_{k=0}^n a_k x^k$ is irreducible if:
1) every $a_n$ is prime;
2) all $a_n$ are pair wise coprime;
3) no other conditions?
I heard problem 2 was asked somewhere but I don't know neither answer nor the source.