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)

use strict;
use warnings;
use v5.18.0;
use Math::BigInt qw(bgcd);
use List::Util qw(sum);

chomp(my $arg = <>);

my @nums;

for (my $a = 1; $a <= $arg; $a++) {
  for (my $b = 1; $b <= $arg; $b++) {
    for (my $c = 1; $c <= $arg; $c++) {
      push(@nums, bgcd($a, $b, $c));
    }
  }
}

print sum(@nums)."\n";