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 Haskell (GHC 8.8.3) (judge verdict: Accepted)

import qualified Data.ByteString.Char8 as BS
import Data.Maybe

permu3 :: Int -> Int -> Int -> Int
permu3 a b c
    | a==b && b==c         = 1
    | a==b || b==c || c==a = 3
    | otherwise            = 6

main :: IO ()
main = do 
    n <- fst . fromJust <$> ( BS.readInt <$> BS.getLine ) :: IO Int
    let t1 = 
            [ permu3 a b c * foldl1 gcd [a,b,c]
            | a <- [1..n], b <- [a..n], c <- [b..n]
            ]
    print $ sum t1