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, a, b, c
    integer :: total = 0
    read *, k

    do a=1, k
        do b = 1, k
            do c = 1, k
                total = total + euclid(a, euclid(c, b))
            enddo
        enddo
    enddo
    print *, total

contains
    integer function euclid(a, b)
        integer, intent(in) :: a,b
        integer bigger, smaller, rem

        if ( a < b ) then
            bigger = b
            smaller = a
        else
            bigger = a
            smaller = b
        endif

        do
            rem = mod(bigger, smaller)
            if ( rem == 0 ) then
                euclid = smaller
                exit
            else
                bigger = smaller
                smaller = rem
            endif
        enddo
    end function
end program main