For some $k>2$, form a sequence whose nth term is the nth prime that is not a divisor of $k$ modulo $k$.
e.g. for $k=4$ the sequence would be 1,3,1,3,3,1,1,3,3,1,3,1...
Is this sequence normal, in the sense that every string of length $w$ consisting of numbers in $\{1..k\}$ coprime to $k$ occurs as a block of consecutive terms with asymptotic density $\frac{1}{\phi(k)^w}$?
Note that the case $w=1$ is equivalent to Dirichlet's theorem.
I don't believe that this is known, although it is conjectured to be true (and would surely follow, for example, from the Hardy-Littlewood prime $k$-tuples conjecture). There has been research into getting constellations of primes in particular residue classes, but it's hard to ensure that you get consecutive primes in the right residue classes. Some important earlier work is by Daniel Shiu, I believe; it's possible, using the recent Maynard/Tao techniques, that somebody might have made some progress.