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)

fn factorize(mut n: u64) -> Vec<u64> {
    let mut f = vec![1];
    for k in 2.. {
        if k * k > n {
            break;
        }
        if n % k == 0 {
            let mut c = 0;
            while n % k == 0 {
                n /= k;
                c += 1;
            }
            let mut next = f.clone();
            let mut mul = 1;
            for _ in 0..c {
                mul *= k;
                for &f in f.iter() {
                    next.push(f * mul);
                }
            }
            f = next;
        }
    }
    if n > 1 {
        let mut next = f.clone();
        for f in f {
            next.push(f * n);
        }
        f = next;
    }
    f
}

fn run() {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).unwrap();
    let n: u64 = s.trim().parse().unwrap();
    let mut ok = vec![];
    for f in factorize(n) {
        if f <= 1 {
            continue;
        }
        let mut n = n;
        while n % f == 0 {
            n /= f;
        }
        if n % f == 1 {
            ok.push(f);
        }
    }
    for f in factorize(n - 1) {
        if f <= 1 {
            continue;
        }
        let mut n = n;
        while n % f == 0 {
            n /= f;
        }
        if n % f == 1 {
            ok.push(f);
        }
    }
    ok.sort();
    ok.dedup();
    println!("{}", ok.len());
}

fn main() {
    run();
}