Prime numbers and Dirichlet's theorem

Question

Let $(a_n)_{n\geq 1}$ and $(b_n)_{n\geq 1}$ be two sequences of positive integers such that $g.c.d(a_n, b_n)=1, \ \forall \ n\geq 1$. For each $n\geq 1$, by Dirichlet's theorem, we know that there is an infinity of primes of the form $a_n\cdot k_n+b_n$. My question is: Is it true that there is a positive constant $k$ (independent of $n$) such that $a_1\cdot k+b_1, a_2\cdot k+b_2, \ldots, a_n\cdot k+b_n$, and so on, are all primes ? I am interested in the following case (even though I don't think it helps much for an answer): $b_n=1, \ \forall \ n\geq 1$, and $a_n=4p_n, \ \forall \ n\geq 1$, where $p_n$ is the $n$th prime number of the form $4\cdot h+3$. Any hint/opinion is appreciated.