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 C++ (Clang 10.0.0) (judge verdict: Accepted)

#include <iostream>
using namespace std;

long long gcd1(long long a, long long b)
{
    if(b == 0) return a;
    return gcd1(b, a%b);
}

long long gcd(long long a, long long b, long long c)
{
    return gcd1(a,gcd1(b,c));
}

int main()
{
    long long k;
    cin >> k;

    long long sum = 0;
    for(long long a = 1; a <= k; ++a)
    {
        for(long long b = 1; b <= k; ++b)
        {
            for(long long c = 1; c <= k; ++c)
            {
                sum += gcd(a,b,c);
            }
        }
    }

    cout << sum << endl;
}