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 (1.42.0) (judge verdict: Accepted)

use std::io::stdin;
use std::str::FromStr;

fn main(){
    let n: usize = get_one();
    if n == 2 { println!("{}", 1); return; }
    let ds = divisors(n);
    let mut ans = 0;
    for d in &ds[1..] {
        let mut m = n;
        while m % d == 0 {
            m = m / d;
        }
        if m % d == 1 { ans += 1; }
    }
    ans += divisors((n as i64 - 1) as usize)[1..].len();
    println!("{}", ans);
}

fn divisors(n: usize) -> Vec<usize> {
    let mut ds = Vec::new();
    for i in 1..n {
        if i * i > n { break; }
        if n % i == 0 {
            ds.push(i);
            let z = n / i;
            if z != i { ds.push(z); }
        }
    }
    return ds;
}

#[allow(dead_code)]
fn get_line() -> String {
    let mut s = String::new();
    match stdin().read_line(&mut s){
        Ok(_) => {s.trim().to_string()}
        Err(_) => String::new()
    }
}

#[allow(dead_code)]
fn get_vec<T: std::str::FromStr>() -> Vec<T> {
    let line = get_line();
    line.split_whitespace().filter_map(|x| x.parse().ok()).collect()
}

#[allow(dead_code)]
fn get_one<T: FromStr + Copy>() -> T {
    let v = get_vec();
    v[0]
}

#[allow(dead_code)]
fn get_pair<T: FromStr + Copy>() -> (T, T) {
    let v = get_vec();
    (v[0], v[1])
}

#[allow(dead_code)]
fn get_triple<T: FromStr + Copy>() -> (T, T, T) {
    let v = get_vec();
    (v[0], v[1], v[2])
}

#[allow(dead_code)]
fn get_chars() -> Vec<char> {
    get_line().chars().collect()
}

#[allow(dead_code)]
fn vec_min(xs: &Vec<i64>) -> i64 {
    xs.iter().map(|&x|x).fold(std::i64::MAX, std::cmp::min)
}

#[allow(dead_code)]
fn vec_max(xs: &Vec<i64>) -> i64 {
    xs.iter().map(|&x|x).fold(std::i64::MIN, std::cmp::max)
}

#[allow(dead_code)]
fn vec_sum(xs: &Vec<i64>) -> i64 {
    xs.iter().fold(0, |acc, &x| acc+x)
}