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 Rust (1.42.0) (judge verdict: Accepted)

use proconio::{input, fastout};
use num::integer::gcd;

#[fastout]
fn main() {
    input! {
        n: u64,
    }
    let mut ans: u64 = 0;

    for i in 1..=n {
        for j in 1..=n {
            let ij = gcd(i, j);
            if ij == 1 {
                ans += n;
                continue;
            }
            for k in 1..=n {
                ans += gcd(ij, k);
            }
        }
    }
    println!("{}", ans);
}