For example the list $(2, 1, 2, 1)$ is congruent $\pmod 3$ to the consecutive primes $(5, 7, 11, 13)$. But how about the list $(1,1,1,1,1,1,1,1,2,3,4,3,2,3,1) \mod 5$?

More generally, we are given some integer $n \geq 2$ and a finite list of integers that are coprime to and less than $n$. Is it always possible to produce the same list by consecutive primes $\pmod n$?

Formally: given $n \geq 2$ and $(a_0,a_1,\cdots \,a_k)$ such that for all $i$, $GCD(a_i, n) = 1$, is there a list of consecutive primes such that each $p_i \equiv a_i \pmod n$?