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

    // AtCoder Beginner Contest 162
    // C - Sum of gcd of Tuples (Easy)

    $max = in_int();

    $sum = 0;
    for ($i = 1; $i <= $max; $i++) {
        for ($j = 1; $j <= $max; $j++) {
            for ($k = 1; $k <= $max; $k++) {
                $sum += gcd($i, gcd($j, $k));
            }
        }
    }

    out($sum);

    function gcd($m, $n) {
        if ($n > $m) {
            $tmp = $n;
            $n = $m;
            $n = $tmp;
        }
        while($n !== 0){
            $tmp = $n;
            $n = $m % $n;
            $m = $tmp;
        }
        return $m;
    }


    // -----------------------------------------------------
    function in() { return trim(fgets(STDIN)); }
    function in_int() { return (int)in(); }
    function in_vals() { return explode(' ', in()); }
    function in_ints() { return array_map('intval', in_vals()); }
    function out($s) { echo $s . PHP_EOL; }
    function yes() { out('Yes'); }
    function no() { out('No'); }