A245663 - OEIS

%I #26 Sep 17 2024 16:00:44

%S 10,43,86,87,188,156,291,364,432,410,7,510,4,4,4,813,4,1079,4,1900,6,

%T 10,6,2330,2147,5,3463,2401,7,2522,5,3884,5,5,8316,3621,5,8,8,4866,5,

%U 5,5,5,6302,5,5,8616,5

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

%C a(n)! > n. - _Robert Israel_, Aug 17 2014

%H Dimitri Zucker, <a href="https://www.youtube.com/watch?v=b2zhTrggRDE">Factorial Fact Frenzy (!)</a>, Combo Class Youtube video (2022).

%e 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.

%e 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.

%p f:= proc(n)

%p   local f,k;

%p   for k from 1 do

%p     f:= k!;

%p     if f mod convert(convert(f,base,n),`+`) <> 0 then return k fi;

%p   od

%p end proc:

%p seq(f(n),n=2..30); # _Robert Israel_, Aug 10 2014

%t 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 *)

%o (Haskell)

%o fac :: Integer -> Integer

%o fac 0 = 1

%o fac n = foldl (*) 1 [2..n]

%o base 0 b = []

%o base a b = (a `mod` b) : base ((a-(a `mod` b)) `div` b) b

%o bAse a b = reverse (base a b)

%o sigbAse a b = foldl (+) 0 (bAse a b)

%o f n = [k | k <- [1..], not ((fac k) `mod` (sigbAse (fac k) n) == 0)] !! 0

%o main = print (map f [2..20]) -- generates values for n = 2 through 20. May be slow for values over 30.

%o (PARI) sumd(k, n) = my(d = digits(k, n)); sum(j=1, #d, d[j]);

%o a(n) = {k = 2; fk = k!; while (fk % sumd(fk, n) == 0, k++; fk = k!); k;} \\ _Michel Marcus_, Aug 10 2014

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

%Y Cf. A066419.

%K nonn,base

%O 2,1

%A _G. H. Faust_, Jul 28 2014