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)

open Printf
open Scanf

let rec gcd a b =
  if b = 0 then a
  else gcd b (a mod b)

let solve n =
  let rec loop i j k sm =
    if i > n then sm
    else if j > n then loop (i + 1) 1 1 sm
    else if k > n then loop i (j + 1) 1 sm
    else loop i j (k + 1) (sm + gcd i (gcd j k)) in
  loop 1 1 1 0
   
let () =
  scanf "%d " solve |> printf "%d\n"