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', $num);
$ans = 0;
for ($i=1; $i <= $num ; $i++) { 
    for ($j=1; $j <= $num ; $j++) {
        $ij = gmp_gcd($i, $j); 
        for ($k=1; $k <= $num ; $k++) { 
            if ($k === 1) {
                $ans++;
                continue;
            }
            $ans+=gmp_gcd($ij, $k); 
        }
    }
}
echo $ans;