Are infinitely many primes $P_{2n-1}$ expressible as $3k-1$? 

The primes $P_{2n-1}$ are every other prime beginning with $2$: [$2,5,11,17,23,31,\cdots$][1]. The first few are of the form $3k-1$, but $31$ is not.


  [1]: https://oeis.org/A031368