Dirichlet's theorem on primes in arithmetic progressions tells us that given the $n$ prime $p_n$ there are infinitely many primes of the form $mp_n + 1$ Obviously, for $p_1 = 2$ the primes of that form are merely the odd primes. For the other primes, $m$ has to be even, but not much else appears obvious.

The leftmost column of this table gives the $n$ prime, the second column from the left gives the smallest prime of the form $mp_n + 1$ the third column from the left gives the second smallest prime of that form, etc. Apart from the leftmost column, none of the columns contain a sequence in ascending order.