cp's OEIS Frontend

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

%I A245663 #26 Sep 17 2024 16:00:44
%S A245663 10,43,86,87,188,156,291,364,432,410,7,510,4,4,4,813,4,1079,4,1900,6,
%T A245663 10,6,2330,2147,5,3463,2401,7,2522,5,3884,5,5,8316,3621,5,8,8,4866,5,
%U A245663 5,5,5,6302,5,5,8616,5
%N A245663 The first number k such that the sum of the base n digits of k! does not divide k!.
%C A245663 a(n)! > n. - _Robert Israel_, Aug 17 2014
%H A245663 Dimitri Zucker, <a href="https://www.youtube.com/watch?v=b2zhTrggRDE">Factorial Fact Frenzy (!)</a>, Combo Class Youtube video (2022).
%e A245663 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 A245663 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 A245663 f:= proc(n)
%p A245663   local f,k;
%p A245663   for k from 1 do
%p A245663     f:= k!;
%p A245663     if f mod convert(convert(f,base,n),`+`) <> 0 then return k fi;
%p A245663   od
%p A245663 end proc:
%p A245663 seq(f(n),n=2..30); # _Robert Israel_, Aug 10 2014
%t A245663 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 A245663 (Haskell)
%o A245663 fac :: Integer -> Integer
%o A245663 fac 0 = 1
%o A245663 fac n = foldl (*) 1 [2..n]
%o A245663 base 0 b = []
%o A245663 base a b = (a `mod` b) : base ((a-(a `mod` b)) `div` b) b
%o A245663 bAse a b = reverse (base a b)
%o A245663 sigbAse a b = foldl (+) 0 (bAse a b)
%o A245663 f n = [k | k <- [1..], not ((fac k) `mod` (sigbAse (fac k) n) == 0)] !! 0
%o A245663 main = print (map f [2..20]) -- generates values for n = 2 through 20. May be slow for values over 30.
%o A245663 (PARI) sumd(k, n) = my(d = digits(k, n)); sum(j=1, #d, d[j]);
%o A245663 a(n) = {k = 2; fk = k!; while (fk % sumd(fk, n) == 0, k++; fk = k!); k;} \\ _Michel Marcus_, Aug 10 2014
%Y A245663 Sum of the base n digits of k for n = 2, 3 and 10 respectively: A000120, A053735, A007953.
%Y A245663 Cf. A066419.
%K A245663 nonn,base
%O A245663 2,1
%A A245663 _G. H. Faust_, Jul 28 2014