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 v5.26; # strict say state biwise
use warnings;
use List::Util qw(reduce any all none notall first max maxstr min minstr product sum sum0 pairs unpairs pairkeys pairvalues pairfirst pairgrep pairmap shuffle uniq uniqnum uniqstr);

chomp (my $k = <STDIN>);
my $ans = 0;
my @cal;
my $gcd = sub {
  my ($x, $y) = @_;
  return $cal[$x][$y] if $cal[$x][$y]>0;
  my ($v, $u) = ($x, $y);
  while ($y) {
    my $r = $x % $y;
    $r += $y if $r < 0;
    $x = $y;
    $y = $r;
  }
  $cal[$v][$u] = $x;
  return $x;
};
for my $i (1..$k) {
  for my $j (1..$k) {
    my $c = $gcd->($i, $j);
    for my $l (1..$k) {
      $ans += $gcd->($c, $l);
    }
  }
}
say $ans;