def factors_slow(n):
    """Count the number of factors of a positive integer n.

    >>> factors_slow(4)
    3
    >>> factors_slow(28)
    6
    """
    total = 0
    i = 1
    while i <= n:
        if n % i == 0:
            total += 1
        i += 1
    return total


def factors_fast(n):
    """Count the number of factors of a positive integer n.

    >>> factors_fast(4)
    3
    >>> factors_fast(28)
    6
    """
    total = 0
    i = 1
    while i * i < n:
        if n % i == 0:
            total += 2
        i += 1
    if i * i == n:
        total += 1
    return total


def fib(n):
    """Return the nth Fibonacci number.

    >>> fib(1)
    1
    >>> fib(8)
    21
    """
    if n == 0 or n == 1:
        return n
    else:
        return fib(n - 2) + fib(n - 1)


def overlap(m, n):
    count = 0
    for i in range(m):
        for j in range(n):
            count += 1
    return count


def fib_iter(n):
    """Return the nth Fibonacci number.

    >>> fib_iter(1)
    1
    >>> fib_iter(8)
    21
    """
    i = 0
    curr, next = 0, 1
    while i < n:
        curr, next = next, curr + next
        i += 1
    return curr


def exp_slow(b, n):
    """Calculate b to the power n where n is a nonnegative integer.

    >>> exp_slow(2, 4)
    16
    >>> exp_slow(2, 10)
    1024
    """
    if n == 0:
        return 1
    else:
        return b * exp_slow(b, n - 1)


sq = lambda x: x * x


def exp_fast(b, n):
    """Calculate b to the power n where n is a nonnegative integer.

    >>> exp_fast(2, 4)
    16
    >>> exp_fast(2, 10)
    1024
    """
    if n == 0:
        return 1
    elif n % 2 == 0:
        return sq(exp_fast(b, n // 2))
    else:
        return b * exp_fast(b, n - 1)


def sum_naturals(n):
    """Calculates 1 + 2 + 3 + ... + n.

    >>> sum_naturals(2)
    3
    >>> sum_naturals(5)
    15
    """
    total = 0
    for i in range(1, n + 1):
        total += i
    return total


def sum_powers_of_two(n):
    """Calculates 1 + 2 + 4 + 8 + ... + 2^n.

    >>> sum_powers_of_two(1)
    3
    >>> sum_powers_of_two(4)
    31
    """
    total = 0
    for i in range(n + 1):
        total += 2 ** i
    return total