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)

;;(declaim (optimize (speed 0) (safety 3) (debug 3)))
#+sbcl (declaim (sb-ext:unmuffle-conditions sb-ext:compiler-note))
(declaim (optimize (speed 3) (safety 0) (debug 0)))

(declaim (type (integer 1 200) *k*))
(defvar *k* (read))

;; gcd(a,b,c) = gcd(gcd(a,b),c)

(princ (loop for a from 1 to *k* summing
	     (loop for b from 1 to *k*
		   for z = (gcd a b) summing
		   (loop for c from 1 to *k* summing
			 (gcd z c) fixnum) fixnum) fixnum))