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)

// This code is generated by [cargo-atcoder](https://github.com/tanakh/cargo-atcoder)
// Original source code:
/*
#![allow(unused_imports)]
#![allow(unused_macros)]
#![allow(non_snake_case)]
#![allow(dead_code)]
#![allow(unused_variables)]
#![allow(unused_mut)]
#![allow(unused_assignments)]
use proconio::input;
use proconio::marker::*;
use std::collections::*;
use std::cmp::*;
use std::mem::swap;
use std::f64::consts::*;

const MOD: u64 = 1000000007;
const INF: usize = std::usize::MAX / 4;

fn divisors(N: u64) -> Vec<u64> {
    let mut ret = vec![];

    let mut d = 1;
    while d * d <= N {
        if N % d == 0 {
            ret.push(d);
            let m = N / d;
            if m != d {
                ret.push(m);
            }
        }
        d += 1;
    }

    return ret;
}

fn main() {
    input! {
        N: u64,
    }

    let mut V = HashSet::new();
    for d in divisors(N) {
        if d == 1 {
            continue;
        }
        let mut M = N;
        while M % d == 0 {
            M /= d;
        }
        M %= d;
        if M == 1 {
            V.insert(d);
        }
    }
    for d in divisors(N - 1) {
        if d == 1 {
            continue;
        }
        let mut M = N;
        while M % d == 0 {
            M /= d;
        }
        M %= d;
        if M == 1 {
            V.insert(d);
        }
    }

    println!("{}", V.len());
}
*/

fn main() {
    let exe = "/tmp/bin11832597";
    std::io::Write::write_all(&mut std::fs::File::create(exe).unwrap(), &decode(BIN)).unwrap();
    std::fs::set_permissions(exe, std::os::unix::fs::PermissionsExt::from_mode(0o755)).unwrap();
    std::process::exit(std::process::Command::new(exe).status().unwrap().code().unwrap())
}

fn decode(v: &str) -> Vec<u8> {
    let mut ret = vec![];
    let mut buf = 0;
    let mut tbl = vec![64; 256];
    for i in 0..64 { tbl[TBL[i] as usize] = i as u8; }
    for (i, c) in v.bytes().filter_map(|c| { let c = tbl[c as usize]; if c < 64 { Some(c) } else { None } }).enumerate() {
        match i % 4 {
            0 => buf = c << 2,
            1 => { ret.push(buf | c >> 4); buf = c << 4; }
            2 => { ret.push(buf | c >> 2); buf = c << 6; }
            3 => ret.push(buf | c),
            _ => unreachable!(),
        }
    }
    ret
}

const TBL: &'static [u8] = b"ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/";
const BIN: &'static str = "<base64-encoded ELF executable removed>";