#!/usr/bin/env python3

from random import randint
from collections import Counter

FACTOR_SIEVE = []

def primegen(n):
    sieve = [False] * n
    yield 2
    for p in range(3, n, 2):
        if sieve[p]:
            continue
        for i in range(p**2, n, 2*p):
            sieve[i] = True
        yield p

def iter_factors(n):
    sieve = init_factors(n)
    for i in range(n):
        yield pfactors(i, sieve)

def init_factors(n):
    sieve = [False] * n
    for p in range(3, n, 2):
        if sieve[p]:
            continue
        for i in range(p**2, n, 2*p):
            sieve[i] = p
    return sieve

def pfactors(n, sieve=FACTOR_SIEVE):
    if n < 1:
        if not n:
            return Counter()
        n = abs(n)
    fctrs = Counter()
    while not n % 2:
        fctrs[2] += 1
        n //= 2
    while sieve[n]:
        fctrs[sieve[n]] += 1
        n //= sieve[n]
    if n != 1:
        fctrs[n] += 1
    return fctrs


def miller_rabin(n, k=10):
    if not (n % 2 and n % 3):
        return False
    d = n - 1
    r = 0
    while not r % 2:
        d //= 2
        r += 1
    for i in range(k):
        a = randint(2, n-2)
        x = pow(a, d, n)
        if x == 1 or x == n - 1:
            continue
        for j in range(r-1):
            x = pow(x, 2, n)
            if x == n - 1:
                break
        else:
            return False
    return True