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 Fortran (GNU Fortran 9.2.1) (judge verdict: Accepted)

program sum_of_gcd_of_tuples_easy
  implicit none
  integer(8) :: k, a, b, c, s = 0_8, g
  read(*,*) k
  do a = 1_8, k
    do b = 1_8, k
      g = gcd(a,b)
      do c = 1_8, k
        s = s+gcd(g,c)
      end do
    end do
  end do
  write(*,'(i0)') s
contains
  integer(8) function gcd(a,b)
    integer(8), intent(in) :: a, b
    integer(8) :: x, y
    x = max(abs(a),abs(b))
    y = min(abs(a),abs(b))
    do while (y > 0)
      gcd = y
      y = mod(x,y)
      x = gcd
    end do
  end
end program sum_of_gcd_of_tuples_easy