a(19) > 250000. All values of this sequence must be odd numbers because an even k would produce 64 * 3^k - 1 that is the difference of two squares, so not prime for any such difference > 4.
Conjecture: This sequence intersects with A387761 at k = 1 and k = 7 to form twin primes with centers N = 2^6 * 3^1 = 192 = A027856(7) and N = 2^6 * 3^7 = 139968 = A027856(11). A covering system can be constructed that eliminates all other intersections (see linked program), and the search up to 250000 makes the probability of another intersection vanishingly small.
