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 main
    implicit none
    integer K, ans, a, b, c
    ans = 0
    read (*, *) K
    do a = 1, K
        do b = 1, K
            do c = 1, K
                ans = ans + gcd(a, gcd(b, c))
            end do
        end do
    end do
    write(*, '(I0)') ans
    stop
contains
    recursive integer function gcd(x, y) result(ret)
        implicit none
        integer x, y
        if (y == 0) then
            ret = x
        else
            ret = gcd(y, mod(x, y))
        end if
        return
    end function
end program main