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
//error_reporting(0);
$N = (int)trim(fgets(STDIN));
//$S = trim(fgets(STDIN));
//list($a,$b,$c) = array_map('intval',explode(" ",trim(fgets(STDIN))));
//$s = trim(fgets(STDIN));
//$a = array_map('intval',explode(" ",trim(fgets(STDIN))));

$c = array();
for($i=1;$i<=$N;$i++) {
  for($j=1;$j<=$N;$j++) {
    for($k=1;$k<=$N;$k++) {
      if (!isset($c[$i][$j])) {
        $c[$i][$j]=gcd($i,$j);
      }
      if (!isset($c[$c[$i][$j]][$k])) {
        $c[$c[$i][$j]][$k]=gcd($c[$i][$j],$k);
      }
      //printf("%d %d %d  %d\n",$i,$j,$k,$s);
      $sum+=$c[$c[$i][$j]][$k];
    }
  }
}
printf("%d\n",$sum);


function gcd($a,$b) {
  if ($b == 0) return $a;
  return gcd($b,$a%$b);
}
function lcm($a,$b) {
  $l = gcd($a,$b);
  return $a / $l * $b;
}

function prime_factors(int $n) {
    $i = 2;
    $factors = [];
    while ($i * $i <= $n) {
        if ($n % $i) $i += 1;
        else {
            $n /= $i;
            $factors[] = $i;
        }
    }
    if ($n > 1) $factors[] = $n;
    return $factors;
}
function is_prime(int $n) : bool {
    $r = true;
    if ($n < 2) $r = false;
    elseif ($n == 2 || $n == 3 || $n == 5) $r = true;
    elseif ($n & 1 == 0) $r = false;
    elseif ($n % 2 == 0 || $n % 3 == 0 || $n % 5 == 0) $r = false;
    else {
        $f = true;
        $v = 7;
        $m = intval($n ** 0.5) + 1;
        while ($v < $m) {
            if ($n % $v == 0) {
                $r = false;
                break;
            }
            $v += $f ? 4 : 2;
            $f = !$f;
        }
    }
    return $r;
}