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 Perl (5.26.1) (judge verdict: Time Limit Exceeded)

use v5.26; # strict say state biwise
use warnings;

chomp (my $k = <STDIN>);
my $ans = 0;
sub gcd {
  my ($x, $y) = @_;
  while ($y) {
    my $r = $x % $y;
    $x = $y;
    $y = $r;
  }
  $x;
}
for my $i (1..$k) {
  for my $j (1..$k) {
    my $c = gcd($i, $j);
    for my $l (1..$k) {
      $ans += gcd($c, $l);
    }
  }
}
say $ans;