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 Common Lisp (SBCL 2.0.3) (judge verdict: Accepted)

(defun Sum-of-gcd-of-Tuples (n)
    (let ((ans 0))
        (progn  
            (loop
                :for i :from 1 :to n
                :do (loop
                        :for j :from 1 :to n
                        :do (loop
                                :for k :from 1 :to n
                                :do (incf ans (gcd (gcd i j) (gcd i k))))))
            ans)))

(princ (Sum-of-gcd-of-Tuples (read)))