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
    //ユークリッドの互除法
    function gcd($a,$b){
        if($b === 0){
            return $a;
        }
        return gcd($b,$a%$b);
    }

    fscanf(STDIN,'%d',$n);

    //参考　⇒　https://note.com/tanon_cp/n/n2eb2bb1d69f9
    //keyがaとbの最大公約数でvalueがその個数になる
    $gcdAAndB = array_fill(1,$n,0);
    for ($a = 1; $a <= $n; ++$a) {
        for ($b = 1; $b <= $n; ++$b) {
            ++$gcdAAndB[gcd($a,$b)];
        }
    }
    //$gcdAAndBとkeyの最大公約数がvalue個ある
    $ans = 0;
    $count = count($gcdAAndB);
    for ($c = 1; $c <= $n; ++$c) {
        for ($index = 1; $index <= $count; ++$index) {
            $ans += gcd($c,$index) * $gcdAAndB[$index];
        }
    }
    echo $ans.PHP_EOL;

?>