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
$K =(int)fgets(STDIN);
$ans = 0;
for ($a = 1; $a <= $K; $a++) {
    for ($b = 1; $b <= $K; $b++) {
        for ($c = 1; $c <= $K; $c++) {
            $ans += gcd($a, gcd($b, $c));
        }
    }
}

echo $ans;

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

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