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 = trim(fgets(STDIN));

$divisors = [];
for ($i=1;$i<=$k;++$i) {
    for ($j=1;$j<=$i;++$j) {
        if ($i%$j === 0) $divisors[$i][] = $j;
    }
}

$count = 0;
for($i=1;$i<=$k;++$i) {
    for($j=$i;$j<=$k;++$j) {
        for($l=$j;$l<=$k;++$l) {
            $common_divisors = array_intersect(
                $divisors[$i],
                $divisors[$j],
                $divisors[$l]
            );

            if ($i === $j || $j === $l || $i === $l) {
                if ($i === $j && $j === $l) {
                    $count += end($common_divisors);
                } else {
                    $count += end($common_divisors) * 3;
                }
            } else {
                $count += end($common_divisors) * 6;
            }
        }
    }
}

echo $count . PHP_EOL;