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 Julia (1.4.0) (judge verdict: Accepted)

using DataStructures

parseInt(x)=parse(Int, x)
parseMap(x::Array{SubString{String},1})=map(parseInt, x)

function main()
    K = readline() |> parseInt
    dd = DefaultDict{Int,Int}(0)
    for i in 1:K
        for j in 1:K
            dd[gcd(i,j)] += 1
        end
    end
    ans = 0
    for key in keys(dd)
        for k in 1:K
           ans += gcd(key,k)*dd[key] 
        end
    end
    println(ans)
end

main()