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: Accepted)

my $N = <STDIN>;

my $sum =0;
my @cal;

for(my $i=1;$i<=$N;$i++){
for(my $j=1;$j<=$N;$j++){
   $s = &gcd($i,$j); 
for(my $k=1;$k<=$N;$k++){
   if( $i==1||$j==1||$k==1){ $sum++;next;}    
   if($s ==1 ){ $sum++;next;}  
   $a = &gcd($k,$s); 
   $sum = $sum + $a;
   #print "gdc($i,$j,$k) $a sum$sum\n";
}
}
}
print "$sum";
exit;

sub gcd{ #最大公約数
  my ($x, $y) = @_;
  return $cal[$x][$y] if $cal[$x][$y]>0;
  my ($v, $u) = ($x, $y);
  while ($y) {
    my $r = $x % $y;
    $r += $y if $r < 0;
    $x = $y;
    $y = $r;
  }
  $cal[$v][$u] = $x;
  return $x;
}