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 = (int)trim(fgets(STDIN));

$list = [];
$sum = 0;
for ($i=1; $i <= $k; $i++) { 
	for ($j=1; $j <= $k; $j++) { 
		for ($m=1; $m <= $k; $m++) { 
			$ary = [$i, $j, $m];
			$sum += gcd($i, gcd($j, $m));
		}
	}
}
echo $sum;


// 最大公約数（ユークリッドの互除法）
function gcd($n1, $n2){
	$large = max($n1, $n2);
	$small = min($n1, $n2);
	if ($small === 1) {
		return 1;
	}
	while($small !== 0){
		$tmp_n = $small;
		$small = $large % $small;
		$large = $tmp_n;
	}
	return $large;
}