Problem:
Score : 300 points

Problem Statement

Find \displaystyle{\sum_{a=1}^{K}\sum_{b=1}^{K}\sum_{c=1}^{K} \gcd(a,b,c)}.

Here \gcd(a,b,c) denotes the greatest common divisor of a, b, and c.

Constraints

1 \leq K \leq 200

K is an integer.

Input

Input is given from Standard Input in the following format:

K

Output

Print the value of \displaystyle{\sum_{a=1}^{K}\sum_{b=1}^{K}\sum_{c=1}^{K} \gcd(a,b,c)}.

Sample Input 1

2

Sample Output 1

9

\gcd(1,1,1)+\gcd(1,1,2)+\gcd(1,2,1)+\gcd(1,2,2)
+\gcd(2,1,1)+\gcd(2,1,2)+\gcd(2,2,1)+\gcd(2,2,2)
=1+1+1+1+1+1+1+2=9

Thus, the answer is 9.

Sample Input 2

200

Sample Output 2

10813692

Solution in OCaml (4.10.0) (judge verdict: Accepted)

let rec gcd a b = if b = 0 then a else gcd b (a mod b)
let ans = ref 0;;
let () =
  Scanf.scanf "%d\n" @@ fun k ->
  for i = k downto 1 do
    for j = k downto 1 do
      for l = k downto 1 do
        ans := (gcd i (gcd j l)) + !ans
      done
    done
  done;
  Printf.printf "%d\n" !ans