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 PHP (7.4.4) (judge verdict: Time Limit Exceeded)

<?php
  $k = intval(fgets(STDIN));
 
  $total = 0;
  $list = [];

  for($i=0; $i<$k; $i++){
    $list[$i+1] = [];
  }

  for($a=1; $a<=$k; $a++){
    for($b=1; $b<=$k; $b++){
      if(isset($list[$a][$b])){
        $tmp = $list[$a][$b];
      }else{
        $tmp = gmp_gcd($a, $b);
        $list[$a][$b] = $tmp;
      }
      
      for($c=1; $c<=$k; $c++){
        if(isset($list[$tmp][$c])){
          $tmp2 = $list[$tmp][$c];
        }else{
          $tmp2 = gmp_gcd($tmp, $c);
          $list[$tmp][$c] = $tmp2;
        }
        $total += $tmp2;
      }
    }
  }
 
  echo $total."\n";