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 n m =
  if m = 0 then n
  else gcd m (n mod m)

let () = Scanf.scanf "%d" @@ fun k ->
  Printf.printf "%d\n" @@
  Array.fold_left (Array.fold_left (Array.fold_left ( + ))) 0 @@
  Array.init k @@ fun a ->
    Array.init k @@ fun b ->
      Array.init k @@ fun c ->
        gcd (a + 1) (gcd (b + 1) (c + 1))