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
  implicit none
  integer :: k, i, j
  integer(8) :: x(200) = 0, y(200) = 0
  read(*,*) k
  do i = k, 1, -1
    x(i) = (k / i) ** 3
    do j = 2 * i, k, i
      x(i) = x(i) - x(j)
    end do
    y(i) = i
  end do
  write(*,'(i0)') dot_product(x, y)
end program sum_of_gcd_of_tuples