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: Accepted)

<?php
fscanf(STDIN, "%d", $n);

$sum = 0;
for ($i=1; $i <= $n; $i++) {
  for ($j=1; $j <= $n; $j++) {
    $ij = gcd($i, $j);
    for ($k=1; $k <= $n; $k++) {
      $sum += gcd($ij, $k);
    }
  }
}

echo $sum;

function gcd($m, $n)
{
  if($n > $m) list($m, $n) = array($n, $m);

  while($n !== 0){
    $tmp_n = $n;
    $n = $m % $n;
    $m = $tmp_n;
  }
  return $m;
}