One could talk of signed integers here: 0, 0, 0, +3, -3, +9, +27, -18, 0, 0, +9, +9, depending on whether the minimum sum-of-digits finds itself above (plus) or below (minus) n!. The problem with so doing is that there might exist some n for which a nonzero minimum distance is both plus and minus.
Zeros indicate where there are solutions in
A228311.
List of solutions:
1! 0 (0, 1)
2! 0 (2)
3! 0 (3, 4)
4! +3 (9, 10, 12, 13)
5! -3 (30)
6! +9 (116)
7! +27 (541, 554)
8! -18 (3154, 3186, 3219)
9! 0 (21966)
10! 0 (176755)
11! +9 (1607130)
12! +9 (16305323)
13! 0 (182624820)
