A245663 The first number k such that the sum of the base n digits of k! does not divide k!.

10, 43, 86, 87, 188, 156, 291, 364, 432, 410, 7, 510, 4, 4, 4, 813, 4, 1079, 4, 1900, 6, 10, 6, 2330, 2147, 5, 3463, 2401, 7, 2522, 5, 3884, 5, 5, 8316, 3621, 5, 8, 8, 4866, 5, 5, 5, 5, 6302, 5, 5, 8616, 5

Offset: 2

Author

G. H. Faust, Jul 28 2014

Keywords

• nonn
• base

Comments

a(n)! > n. - Robert Israel, Aug 17 2014

Examples

			The sum of the base-2 digits of 10! is 1+1+0+1+1+1+0+1+0+1+1+1+1+1+0+0+0+0+0+0+0+0=11, which does not divide 10!.  Since the sum of the base-2 digits of k! divides k! for 0 <= k <= 9, a(2) = 10.
The sum of the base-3 digits of 43! is 106, which does not divide 43!.  Since the sum of the base-3 digits of k! divides k! for 0 <= k <= 42, a(3) = 43.
		

Links

• Dimitri Zucker, Factorial Fact Frenzy (!), Combo Class Youtube video (2022).

Crossrefs

Sum of the base n digits of k for n = 2, 3 and 10 respectively: A000120, A053735, A007953.

Cf. A066419.

Programs

Haskell

fac :: Integer -> Integer
fac 0 = 1
fac n = foldl (*) 1 [2..n]
base 0 b = []
base a b = (a `mod` b) : base ((a-(a `mod` b)) `div` b) b
bAse a b = reverse (base a b)
sigbAse a b = foldl (+) 0 (bAse a b)
f n = [k | k <- [1..], not ((fac k) `mod` (sigbAse (fac k) n) == 0)] !! 0
main = print (map f [2..20]) -- generates values for n = 2 through 20. May be slow for values over 30.

Maple

f:= proc(n)
  local f,k;
  for k from 1 do
    f:= k!;
    if f mod convert(convert(f,base,n),`+`) <> 0 then return k fi;
  od
end proc:
seq(f(n),n=2..30); # Robert Israel, Aug 10 2014

Mathematica

a245663[n_Integer] := Module[{f = 2, k = 2}, While[Divisible[f, Total[IntegerDigits[f, n]]] == True, k++; f = k!]; k]; a245663 /@ Range[2, 50] (* Michael De Vlieger, Aug 15 2014 *)

PARI

sumd(k, n) = my(d = digits(k, n)); sum(j=1, #d, d[j]);
a(n) = {k = 2; fk = k!; while (fk % sumd(fk, n) == 0, k++; fk = k!); k;} \\ Michel Marcus, Aug 10 2014