This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A350211 #21 Dec 21 2021 06:34:31
%S A350211 0,1,2,3,4,5,6,12,26,28,32,59,262,391,533,579
%N A350211 Numbers k such that the arithmetic mean of the digits of k! is an integer.
%C A350211 A heuristic argument suggests that this short list is complete. By Stirling's approximation, n! has order n*log(n) digits of which n/4 are terminal zeros. If the remaining digits are random, the mean will be just below 4.5. For n > 6, n! and also its digits sum are divisible by 9. 12! is the only factorial with 9 digits. The others have 27, 30, 36, 81, 522, 846, 1224, and 1350 digits, respectively.
%e A350211 4 is a term because 4! = 24 and (2+4)/2 = 3 is an integer.
%p A350211 q:= n-> (f-> (add(i, i=convert(f, base, 10))/length(f))::integer)(n!):
%p A350211 select(q, [$0..1000])[]; # _Alois P. Heinz_, Dec 19 2021
%t A350211 Do[If[IntegerQ[Mean[IntegerDigits[n!]]], Print[n, " ", Mean[IntegerDigits[n!]]]], {n, 1, 100000}]
%o A350211 (PARI) isok(k) = my(d=digits(k!)); (vecsum(d) % #d) == 0; \\ _Michel Marcus_, Dec 19 2021
%Y A350211 Cf. A000142, A004152, A034886, A058814, A061383.
%K A350211 nonn,base,more
%O A350211 1,3
%A A350211 _Zachary M Franco_, Dec 19 2021