from math import floor, log10

def count_digits_log(n):
    """Return the number of digits of an integer n using logarithms."""
    if n == 0:
        return 1
    return floor(log10(abs(n))) + 1

def count_digits_str(n):
    """Return the number of digits of an integer n using string len."""
    return len(str(abs(n)))

def count_digits_loop(n):
    """Return the number of digits of an integer n using repeated division."""
    if n == 0:
        return 1
    n = abs(n)
    res = 0
    while n > 0:
        res += 1
        n //= 10
    return res

n = int(input("n = "))
print("count_digits_log({}) =  {}".format(n, count_digits_log(n)))
print("count_digits_str({}) =  {}".format(n, count_digits_str(n)))
print("count_digits_loop({}) = {}".format(n, count_digits_loop(n)))

def is_prime_def(n):
    """Returns `True` if an integer `n` is a prime and `False` otherwise."""
    if n < 2:
        return False
    for d in range(2, n):
        if n % d == 0:
            return False
    return True

def is_prime_sqrt(n):
    """Returns `True` if an integer `n` is a prime and `False` otherwise."""
    if n < 2:
        return False
    d = 2
    while d*d <= n:
        if n % d == 0:
            return False
        d += 1
    return True

n = int(input("n = "))
bool2text = { False: "is not", True: "is" }
print("is_prime_def says that number {} {} a prime.".format(n, bool2text[is_prime_def(n)]))
print("is_prime_sqrt says that number {} {} a prime.".format(n, bool2text[is_prime_sqrt(n)]))

def is_prime_def(n):
    """Returns `True` if an integer `n` is a prime and `False` otherwise."""
    if n < 2:
        return False
    for d in range(2, n):
        if n % d == 0:
            return False
    return True

def is_prime_sqrt(n):
    """Returns `True` if an integer `n` is a prime and `False` otherwise."""
    if n < 2:
        return False
    d = 2
    while d*d <= n:
        if n % d == 0:
            return False
        d += 1
    return True

n = int(input("n = "))
bool2text = { False: "is not", True: "is" }
print("is_prime_def says that number {} {} a prime.".format(n, bool2text[is_prime_def(n)]))
print("is_prime_sqrt says that number {} {} a prime.".format(n, bool2text[is_prime_sqrt(n)]))

from math import floor, sqrt

def is_prime_sqrt(n):
    """Returns `True` if an integer `n` is a prime and `False` otherwise."""
    if n < 2:
        return False
    for d in range(2, floor(sqrt(n)) + 1):
        if n % d == 0:
            return False
        d += 1
    return True

n = int(input("n = "))
bool2text = { False: "is not", True: "is" }
print("is_prime_sqrt says that number {} {} a prime.".format(n, bool2text[is_prime_sqrt(n)]))

from math import floor, sqrt

def is_prime_py(n):
    """Returns `True` if an integer `n` is a prime and `False` otherwise."""
    return (n >= 2) and all(n%d != 0 for d in range(2, floor(sqrt(n)) + 1))

n = int(input("n = "))
bool2text = { False: "is not", True: "is" }
print("is_prime_py says that number {} {} a prime.".format(n, bool2text[is_prime_py(n)]))

from time import time

def divisors_sum(n):
    """Returns the sum of all natural divisiors of a natural number `n` except `n` itself."""
    res = 0
    for d in range(1, n):
        if n % d == 0:
            res += d
    return res

# More Pythonic version
def divisors_sum_py(n):
    """Returns the sum of all natural divisiors of a natural number `n` except `n` itself."""
    return sum(d for d in range(1, n) if n % d == 0)

n = int(input("n = "))
st = time()
for a in range(1, n+1):
    for b in range(a+1, n+1):
        if a == divisors_sum(b) and b == divisors_sum(a):
            print((a, b))
print("Done in {:.6f} seconds.".format(time() - st))

n = int(input("n = "))
st = time()
for a in range(1, n+1):
    b = divisors_sum(a)
    if a < b and a == divisors_sum(b):
        print((a, b))
print("Done in {:.6f} seconds.".format(time() - st))

from time import time
from math import floor, sqrt

def is_prime_sqrt(n):
    """Returns `True` if an integer `n` is a prime and `False` otherwise."""
    if n < 2:
        return False
    for d in range(2, floor(sqrt(n)) + 1):
        if n % d == 0:
            return False
        d += 1
    return True

def print_prime_factors_def(n):
    """Prints all the prime factors of `n`, each of them once."""
    for p in range(2, abs(n)+1):
        if n % p == 0 and is_prime_sqrt(p):
            print(p)

n = int(input("n = "))
print("The prime factors of {} are:".format(n))
st = time()
print_prime_factors_def(n)
print("Total time:", time() - st)

from time import time

def print_prime_factors_div(n):
    """Prints all the prime factors of `n`, each of them once."""
    p = 2
    n = abs(n)
    while n > 1:
        if n % p == 0:
            print(p)
            while n % p == 0:
                n //= p
        p += 1

n = int(input("n = "))
print("The prime factors of {} are:".format(n))
st = time()
print_prime_factors_div(n)
print("Total time:", time() - st)

from time import time
from math import floor

def print_prime_factors_div(n):
    """Prints all the prime factors of `n`, each of them once."""
    p = 2
    n = abs(n)
    n2 = floor(sqrt(n))
    while p <= n2:
        if n % p == 0:
            print(p)
            while n % p == 0:
                n //= p
            n2 = floor(sqrt(n))
        p += 1
    if n > 1:
        print(n)

n = int(input("n = "))
print("The prime factors of {} are:".format(n))
st = time()
print_prime_factors_div(n)
print("Total time:", time() - st)

def print_prime_factors_div(n):
    """Prints all the prime factors of `n`, each of them
    repeated according to its multiplicity."""
    p = 2
    n = abs(n)
    while n > 1:
        if n % p == 0:
            while n % p == 0:
                print(p)
                n //= p
        p += 1

n = int(input("n = "))
print("The prime factors of {} are:".format(n))
print_prime_factors_div(n)

def print_prime_factors_div(n):
    """Prints all the prime factors of `n`, each of them
    repeated according to its multiplicity."""
    p = 2
    n = abs(n)
    while n > 1:
        while n % p == 0:
            print(p)
            n //= p
        p += 1

n = int(input("n = "))
print("The prime factors of {} are:".format(n))
print_prime_factors_div(n)

def print_prime_factors_div(n):
    """Prints the summands of the prime factorization of `n`."""
    p = 2
    n = abs(n)
    while n > 1:
        if n % p == 0:
            i = 0
            while n % p == 0:
                i += 1
                n //= p
            print("{}**{}".format(p, i))
        p += 1

n = int(input("n = "))
print("The prime factors of {} are:".format(n))
print_prime_factors_div(n)

def prime_factors_div(n):
    """Returns the generator that makes an iterator for
    the prime factorization of `n`."""
    p = 2
    n = abs(n)
    while n > 1:
        if n % p == 0:
            i = 0
            while n % p == 0:
                i += 1
                n //= p
            yield "{}**{}".format(p, i)
        p += 1

n = int(input("n = "))
print(n, "=", "+".join(prime_factors_div(n)))