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
// Your code here!
$sum = 0;
$stdin = trim(fgets(STDIN));  
for ($a=1;$a<=$stdin;$a++){
    for ($b=1;$b<=$stdin;$b++){
        for ($c=1;$c<=$stdin;$c++){
            if($a ==1 || $b==1 || $c==1){
                $sum += 1;
            }else{
                $sum+=gmp_gcd($c,gmp_gcd($a,$b));
            };
};
};
};
echo $sum;