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A229024 a(n) is the minimum distance to n! for the sum-of-digits of any factorial.

%I A229024 #17 May 17 2014 04:02:36
%S A229024 0,0,0,3,3,9,27,18,0,0,9,9,0
%N A229024 a(n) is the minimum distance to n! for the sum-of-digits of any factorial.
%C A229024 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.
%C A229024 Zeros indicate where there are solutions in A228311.
%C A229024 List of solutions:
%C A229024 1!    0  (0, 1)
%C A229024 2!    0  (2)
%C A229024 3!    0  (3, 4)
%C A229024 4!   +3  (9, 10, 12, 13)
%C A229024 5!   -3  (30)
%C A229024 6!   +9  (116)
%C A229024 7!  +27  (541, 554)
%C A229024 8!  -18  (3154, 3186, 3219)
%C A229024 9!    0  (21966)
%C A229024 10!   0  (176755)
%C A229024 11!  +9  (1607130)
%C A229024 12!  +9  (16305323)
%C A229024 13!   0  (182624820)
%H A229024 Hans Havermann, <a href="http://chesswanks.com/num/a229024(11).txt">Determination of a(11)</a>
%H A229024 Hans Havermann, <a href="http://chesswanks.com/num/a229024(12).txt">Determination of a(12)</a>
%H A229024 Hans Havermann, <a href="http://chesswanks.com/num/a229024(13).txt">Determination of a(13)</a>
%e A229024 The minimum distance to 4! is 3, given by the sum of digits for 9!, 10!, 12!, or 13!.
%e A229024 The minimum distance to 5! is also 3, given by the sum of digits of 30!.
%Y A229024 Cf. A004152, A228311.
%K A229024 nonn,base,hard
%O A229024 1,4
%A A229024 _Hans Havermann_, Sep 11 2013
%E A229024 a(13) from _Hans Havermann_, Nov 04 2013