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: Accepted)

use std::io;
use std::cmp;
use std::collections::BinaryHeap;
use std::collections::HashMap;
use std::collections::HashSet;
use std::f64::consts::PI;

fn read_line() -> String {
    let mut buf = String::new();
    io::stdin().read_line(&mut buf).ok();
    buf
}

fn func1(n: u64) -> u64 {
    if n == 2 {
        return 0;
    }
    let mut res = 1;
    for k in 2..cmp::min(n-1, 2_000_010) {
        if k * k > n-1 {
            break;
        }
        if (n-1) % k == 0 {
            if (n-1) / k == k {
                res += 1;
            } else {
                res += 2;
            }
        }
    }
    res
}

fn func2(n: u64) -> u64 {
    let mut res = 1;
    for k in 2..cmp::min(n, 2_000_010) {
        if n % k == 0 {
            let mut tmp = k;
            while n % tmp == 0 {
                if (n / tmp) % k == 1 {
                    res += 1;
                    break;
                }
                tmp *= k;
            }
        }
    }
    res
}


fn main() {
    let line: Vec<u64> = read_line().split_whitespace().map(|s| s.parse().unwrap()).collect();
    let n = line[0];
    println!("{}", func1(n) + func2(n));
}

#[cfg(test)]
mod test {
    fn greedy(n: u64) -> u64 {
        let mut res = 0;
        for i in 2..n+1 {
            let mut tmp = n;
            while tmp >= i {
                if tmp % i == 0 {
                    tmp /= i;
                } else {
                    tmp -= i;
                }
            }
            if tmp == 1 {
                res += 1;
            }
            
        }
        res
    }
    use func1;
    use func2;

    #[test]
    fn test() {
        for i in 10000..50000 {
            let ans = greedy(i);
            println!("testing :{}, ans: {}", i, ans);
            assert_eq!(ans, func1(i) + func2(i));
        }
    }
}