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)

my $N = <STDIN>;

my $sum =0;

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 ($a,$b) = @_;
    if($b == 0){
        return $a;
    }
   
    return gcd($b, $a%$b);
}