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 $N = <STDIN>;
if($N==1){print "$N";exit;}
my $sum =0;
my $mod = 10**9+7;
my %del;
for(my $i=$N;$i>0;$i--){
 $count = int($N /$i);
 $del{$i} = powmod($count,3,$mod);
}
for(my $i=$N;$i>1;$i--){
 $del{1}=$del{1} - $del{$i};
 for(my $j=2;$j<=sqrt($i);$j++){
    if( ($i%$j == 0) ){
        $del{$j}= $del{$j} - $del{$i};
        if( $j ==($i/$j)){next;}
        $del{$i/$j}= $del{$i/$j} - $del{$i};
    }
 }
 $sum = $sum + $i*$del{$i};
 #print "i$i del$del{$i} $sum\n";
}
$sum = $sum + $del{1};
print "$sum";
exit;

sub powmod{
   my ($a,$b,$m) = @_;
   if($b==1){return $a;}
   my $c = powmod(($a*$a)%$m,int($b/2),$m);
   if($b%2){
     $c = ($c*$a)%$m;
   }
   return $c;
}