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)

object Main extends App {
  val in = new java.util.Scanner(System.in)
  val K = in.nextInt
  def gcd(a: Int, b: Int): Int = if (b == 0) a else gcd(b, a % b)
  def gcd(a: Int, b: Int, c: Int): Int = gcd(gcd(a, b), c)

  val gcds = for {
    a <- 1 to K
    b <- 1 to K
    c <- 1 to K
  } yield gcd(a, b, c)

  println(gcds.sum)
}