Indian National Olympiad in Informatics 2015
A string is any nonempty sequence of 0s and 1s. Examples of strings are 00, 101, 111000, 1, 0, 01. The length of a string is the number of symbols in it. For example, the length of 111000 is 6. If u and v are strings, then uv is the string obtained by concatenating u and v. For example if u = 110 and v = 0010 then uv = 1100010.
A string w is periodic if there exists a string v such that w = vn = vv · · · v (n times), for some n ≥ 2. Note that in this case the length of v is strictly less than that of w. For example, 110110 is periodic, because it is vv for v = 110.
Given a positive integer N , find the number of strings of length N which are not periodic. Report the answer modulo M . The non-periodic strings of length 2 are 10 and 01. The non- periodic strings of length 3 are 001, 010, 011, 100, 101, and 110.

-----Input format-----
A single line, with two space-separated integers, N and M .

-----Output format-----
A single integer, the number of non-periodic strings of length N , modulo M .

-----Test Data-----
In all subtasks, 2 ≤ M ≤ 108. The testdata is grouped into 4 subtasks.
Subtask 1 (10 marks) 1 ≤ N ≤ 4000. N is the product of two distinct prime numbers.
Subtask 2 (20 marks) 1 ≤ N ≤ 4000. N is a power of a prime number.
Subtask 3 (35 marks) 1 ≤ N ≤ 4000.
Subtask 4 (35 marks) 1 ≤ N ≤ 150000.

-----Example-----
Here is the sample input and output corresponding to the example above:

-----Sample input-----
3 176

-----Sample output-----
6

Note: Your program should not print anything other than what is specified in the output format. Please remove all diagnostic print statements before making your final submission. A program with extraneous output will be treated as incorrect!

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

ns = {"__name__": "__main__"}
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = "import random\ndef miller_rabin(n, k):\n\n    if n == 2 or n==3:\n        return True\n\n    if n % 2 == 0:\n        return False\n\n    r, s = 0, n - 1\n    while s % 2 == 0:\n        r += 1\n        s //= 2\n    for _ in range(k):\n        a = random.randrange(2, n - 1)\n        x = pow(a, s, n)\n        if x == 1 or x == n - 1:\n            continue\n        for _ in range(r - 1):\n            x = pow(x, 2, n)\n            if x == n - 1:\n                break\n        else:\n            return False\n    return True\npows = {1:2}\nn, k = list(map(int, input().split(' ')))\nT = pow(2,n)\nif miller_rabin(n,6):\n    print((T-2)%k)\nelse:\n    m = 0\n    for i in range(2,n//2+1):\n        if n%i==0:\n            if i>50000:\n                a = 6\n            elif i>20000:\n                a = 5\n            else:\n                a = 4\n            if miller_rabin(i,a):\n                pows[i]=pow(2,i)-2\n                m+=pows[i]\n            else:\n                w = 0\n                for j in pows:\n                    if i%j==0:\n                        w+=pows[j]\n                    if j>i//2:\n                        break\n                pows[i] = pow(2,i)-w\n                m+=pows[i]\n\n    print((T-m-2)%k)\n"
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

```

Measured memory_bytes: 13687