Problem:
Score : 1000 points

Problem Statement

In a two-dimensional plane, we have a rectangle R whose vertices are (0,0), (W,0), (0,H), and (W,H), where W and H are positive integers.
Here, find the number of triangles \Delta in the plane that satisfy all of the following conditions:

Each vertex of \Delta is a grid point, that is, has integer x- and y-coordinates.

\Delta and R shares no vertex.

Each vertex of \Delta lies on the perimeter of R, and all the vertices belong to different sides of R.

\Delta contains at most K grid points strictly within itself (excluding its perimeter and vertices).

Constraints

1 \leq W \leq 10^5

1 \leq H \leq 10^5

0 \leq K \leq 10^5

Input

Input is given from Standard Input in the following format:

W H K

Output

Print the answer.

Sample Input 1

2 3 1

Sample Output 1

12

For example, the triangle with the vertices (1,0), (0,2), and (2,2) contains just one grid point within itself and thus satisfies the condition.

Sample Input 2

5 4 5

Sample Output 2

132

Sample Input 3

100 100 1000

Sample Output 3

461316

Solution in Rust (1.42.0) (judge verdict: Accepted)

#[allow(unused_imports)]
use std::cmp::*;
#[allow(unused_imports)]
use std::collections::*;
use std::io::{Write, BufWriter};
// https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8
macro_rules! input {
    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        let mut next = move || -> String{
            bytes
                .by_ref()
                .map(|r|r.unwrap() as char)
                .skip_while(|c|c.is_whitespace())
                .take_while(|c|!c.is_whitespace())
                .collect()
        };
        input_inner!{next, $($r)*}
    };
}

macro_rules! input_inner {
    ($next:expr) => {};
    ($next:expr, ) => {};
    ($next:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
}

macro_rules! read_value {
    ($next:expr, [graph1; $len:expr]) => {{
        let mut g = vec![vec![]; $len];
        let ab = read_value!($next, [(usize1, usize1)]);
        for (a, b) in ab {
            g[a].push(b);
            g[b].push(a);
        }
        g
    }};
    ($next:expr, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };
    ($next:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    };
    ($next:expr, chars) => {
        read_value!($next, String).chars().collect::<Vec<char>>()
    };
    ($next:expr, usize1) => (read_value!($next, usize) - 1);
    ($next:expr, [ $t:tt ]) => {{
        let len = read_value!($next, usize);
        read_value!($next, [$t; len])
    }};
    ($next:expr, $t:ty) => ($next().parse::<$t>().expect("Parse error"));
}

#[allow(unused)]
macro_rules! debug {
    ($($format:tt)*) => (write!(std::io::stderr(), $($format)*).unwrap());
}
#[allow(unused)]
macro_rules! debugln {
    ($($format:tt)*) => (writeln!(std::io::stderr(), $($format)*).unwrap());
}

fn gcd(x: i64, y: i64) -> i64 {
    if y == 0 {
        x
    } else {
        gcd(y, x % y)
    }
}

fn calc(w: i64, h: i64, k: i64) -> i64 {
    let mut ans: i64 = 0;
    for b in 1..h {
        let lim = 2 * k - 2 + w + b + min(h, w);
        for c in 1..=min(w - 1, lim / b) {
            let det = 2 * k - 2 + gcd(w - c, b);
            let lim = det + c + w;
            let sup = min(h - 1, (lim - b * c) / (w - c));
            let inf = min(max(sup, 0), max(0, (det - b * c) / (w - c)));
            ans += inf;
            for a in inf + 1..=sup {
                if a * (w - c) + b * c <= det + gcd(a, c)
                     + gcd(w, (a - b).abs()) {
                        ans += 1;
                    }
            }
        }
    }
    ans
}

fn solve() {
    let out = std::io::stdout();
    let mut out = BufWriter::new(out.lock());
    macro_rules! puts {
        ($($format:tt)*) => (let _ = write!(out,$($format)*););
    }
    input! {
        w: i64, h: i64, k: i64,
    }
    puts!("{}\n", 2 * (calc(w, h, k) + calc(h, w, k)));
}

fn main() {
    // In order to avoid potential stack overflow, spawn a new thread.
    let stack_size = 104_857_600; // 100 MB
    let thd = std::thread::Builder::new().stack_size(stack_size);
    thd.spawn(|| solve()).unwrap().join().unwrap();
}