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)

use strict;
use warnings;
use utf8;

sub gcd {
    my ($a, $b) = @_;
    return ($b == 0) ? $a : gcd($b, $a % $b);
}

my $k = <STDIN>;
my $sum = 0;

my @first = ();
for my $i (1..$k) {
    my $second = [];
    push @first, $second;
}

for my $a (1..$k) {
    for my $b (1..$k) {
        my $temp = ($first[$a][$b]) ? $first[$a][$b] : gcd($a, $b);
        $first[$a][$b] = $temp;
        $first[$b][$a] = $temp;
        for my $c (1..$k) {
            my $temp2 = ($first[$temp][$c]) ? $first[$temp][$c] : gcd($temp, $c);
            $first[$temp][$c] = $temp2;
            $first[$c][$temp] = $temp2;
            $sum += $temp2;
        }
    }
}

print $sum;