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 Python (3.8.2) (judge verdict: Time Limit Exceeded)

import math

K = int(input())


def gcd3(a, b, c):
    d = math.gcd(a, b)
    e = math.gcd(d, c)
    return e


ans = 0
for x in range(1, K + 1, 1):
    for y in range(1, K + 1, 1):
        tmp = list(map(lambda z: gcd3(x, y, z), range(1, K+1, 1)))
        ans = ans + sum(tmp)

print(ans)