Problem:
Score : 600 points

Problem Statement

Given is a positive integer N.

We will choose an integer K between 2 and N (inclusive), then we will repeat the operation below until N becomes less than K.

Operation: if K divides N, replace N with N/K; otherwise, replace N with N-K.

In how many choices of K will N become 1 in the end?

Constraints

2 \leq N \leq 10^{12}

N is an integer.

Input

Input is given from Standard Input in the following format:

N

Output

Print the number of choices of K in which N becomes 1 in the end.

Sample Input 1

6

Sample Output 1

3

There are three choices of K in which N becomes 1 in the end: 2, 5, and 6.

In each of these choices, N will change as follows:

When K=2: 6 \to 3 \to 1

When K=5: 6 \to 1

When K=6: 6 \to 1

Sample Input 2

3141

Sample Output 2

13

Sample Input 3

314159265358

Sample Output 3

9

Solution in Rust (1.15.1) (judge verdict: Wrong Answer)

fn factorize(a: u64) -> Vec<u64> {
    let u: u64 = (a as f64).sqrt().ceil() as u64;
    let mut res: Vec<u64> = Vec::<u64>::new();
    for i in 2..u {
        if a % i == 0 {
            res.push(i);
            res.push(a / i);
        }
    }
    res.push(a);
    res.sort();
    res
}
fn main() {
    let n: u64 = read();
    let mut n_fac = factorize(n);
    let mut nm1_fac = factorize(n - 1);
    let mut ans = nm1_fac.len() + n_fac.len();
    for nf in n_fac {
        let mut tmp = n;
        while tmp % nf == 0 {
            tmp /= nf;
        }
        if tmp % nf != 1 {
            ans -= 1;
        }
    }
    println!("{}", ans);
}

// =========
#[allow(unused_imports)]
use std::cmp::{max, min};
#[allow(unused_imports)]
use std::collections::{HashMap, HashSet};
#[allow(unused_imports)]
use std::process::exit;

#[allow(dead_code)]
const MOD: usize = 1000000007;

fn read<T: std::str::FromStr>() -> T {
    use std::io::Read;
    let stdin = std::io::stdin();
    let stdin = stdin.lock();
    let token: String = stdin
        .bytes()
        .map(|c| c.expect("failed to read char") as char)
        .skip_while(|c| c.is_whitespace())
        .take_while(|c| !c.is_whitespace())
        .collect();
    token.parse().ok().expect("failed to parse token")
}

// =========