Let $M$ be a natural number $M>1$. For  every prime $p_i$ not dividig $M$ take an arithmetic progression $A_i=k_i+np_i$  , $n\geq 0$ such that $k_i>p_i^2/M$. Is there any $M$ and some choice of the $k_i's$ such that  $\cup A_i= \mathbb{N}$ ?

- MOTIVATION: it is not hard to see that if there is not such a choice for any $M$ Dirichlet's theorem in arithmetic progressions is an easy consequence.

-Is there any counterexample or a proof considering that we can compute the "growth function" of the covers...?