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 Scala (2.13.1) (judge verdict: Accepted)

import scala.io.StdIn._
import scala.annotation.tailrec

object Main extends App {

  @tailrec
  def gcd(a: Int, b: Int): Int = {
    if(b == 0) a else gcd(b, a % b)
  }

  val all = 1 to readInt()

  println(Seq(all.sum,
    all.combinations(2).map(combi => gcd(combi(0), combi(1))).sum*6,
    all.combinations(3).map(combi => gcd(gcd(combi(0), combi(1)), combi(2))).sum*6).foldLeft(0)(_+_))
}