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
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
Programs
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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.
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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
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