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 $\displaystyle{\sum_{a=1}^{K}\sum_{b=1}^{K}\sum_{c=1}^{K} \gcd(a,b,c)}$.

2

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 $9$.

200

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