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 Lua (5.3.2) (judge verdict: Accepted)

local msq = math.sqrt
local mfl, mce = math.floor, math.ceil
local mmi, mma = math.min, math.max
local n = io.read("*n")
-- n = k^a(1-kb)
local ret = 0
local lim = mmi(n - 1, mce(msq(n)))
-- a = 0
for i = 1, lim do
  if n - 1 < i * i then break
  elseif n - 1 == i * i then
    ret = ret + 1
  elseif (n - 1) % i == 0 then
    ret = ret + 2
  end
end
-- a = 1, ..., 40
for a = 1, 40 do
  for k = 2, lim do
    local kpow = 1
    for i = 1, a do
      kpow = kpow * k
    end
    if n < kpow then break end
    if n % kpow == 0 then
      local v = n // kpow
      if v % k == 1 then
        ret = ret + 1
      end
    end
  end
end
print(ret)