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: Wrong Answer)

my $N = <STDIN>;
if($N==1){print "$N";exit;}
my $sum =0;
my %del;
for(my $i=$N;$i>0;$i--){
 $count = int($N /$i);
 $del{$i} = ($count*$count*$count);
}
for(my $i=$N;$i>1;$i--){
  $del{1}=$del{1} - $del{$i};
  $count = int($N /$i);
  for(my $j=2;$j<=$count;$j++){
    $del{$i}= $del{$i} - $del{$i*$j};
 }
 $sum = $sum + $i*$del{$i};
 #print "i$i del$del{$i} $sum\n";
}
$sum = $sum + $del{1};
print "$sum";
exit;