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

1,3

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.

Links

Table of n, a(n) for n=1..16.

Example

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

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}]

Prog

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

Crossrefs

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

Sequence in context: A218345 A059460 A125141 * A282505 A165304 A266642

Adjacent sequences: A350208 A350209 A350210 * A350212 A350213 A350214

Keyword

nonn,base,more

Author

Zachary M Franco, Dec 19 2021

Status

approved