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 Go (1.14.1) (judge verdict: Accepted)

package main

import (
	"bufio"
	"fmt"
	"os"
	"strconv"
)

func main() {
	nextReader = newScanner()
	k := nextInt()
	ans := 0
	for a := 1; a <= k; a++ {
		for b := 1; b <= k; b++ {
			for c := 1; c <= k; c++ {
				ans += gcd(a, gcd(b, c))
			}
		}
	}
	fmt.Println(ans)
}

func gcd(x, y int) int {
	if y == 0 {
		return x
	}
	return gcd(y, x%y)
}

// ---------------------------------------------------------------
// I/O
// ---------------------------------------------------------------
var nextReader func() string

func newScanner() func() string {
	sc := bufio.NewScanner(os.Stdin)
	sc.Buffer(make([]byte, 1024), int(1e11))
	sc.Split(bufio.ScanWords)
	return func() string {
		sc.Scan()
		return sc.Text()
	}
}

func nextString() string { return nextReader() }

func nextInt() int { n, _ := strconv.Atoi(nextReader()); return n }

func nextInts(size int) []int {
	ns := make([]int, size)
	for i := 0; i < size; i++ {
		ns[i] = nextInt()
	}
	return ns
}