A350211 Numbers k such that the arithmetic mean of the digits of k! is an integer.

0, 1, 2, 3, 4, 5, 6, 12, 26, 28, 32, 59, 262, 391, 533, 579

Offset: 1

Author

Zachary M Franco, Dec 19 2021

Keywords

• nonn
• base
• more

Comments

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.

Examples

			4 is a term because 4! = 24 and (2+4)/2 = 3 is an integer.
		

Crossrefs

Cf. A000142, A004152, A034886, A058814, A061383.

Programs

Maple

q:= n-> (f-> (add(i, i=convert(f, base, 10))/length(f))::integer)(n!):
select(q, [$0..1000])[];  # Alois P. Heinz, Dec 19 2021

Mathematica

Do[If[IntegerQ[Mean[IntegerDigits[n!]]], Print[n, " ", Mean[IntegerDigits[n!]]]], {n, 1, 100000}]

PARI

isok(k) = my(d=digits(k!)); (vecsum(d) % #d) == 0; \\ Michel Marcus, Dec 19 2021