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 OCaml (4.02.3) (judge verdict: Wrong Answer)

let rec aux n k =
  if n mod k = 0
  then aux (n / k) k
  else n

let rec solve n k ans =
  if n < k * k
  then ans
  else
    solve n (k + 1) @@
    ans +
    if n mod k = 0
    then (if aux n k mod k = 1 then 1 else 0) + (if aux n (n / k) mod (n / k) = 1 then 1 else 0)
    else if (n - 1) mod k = 0
    then 1 + (if n - 1 = k * k then 0 else 1)
    else 0
let solve n = solve n 2 2

let () = Printf.printf "%d\n" @@ Scanf.scanf "%d" solve