Activity - Python 101

Exercise 1

An approximation of \(\pi\):

In the 17th and 18th centuries, James Gregory and Gottfried Leibniz discovered an infinite series that can be used to calculate \(\pi\):

\[\pi = 4(1-\dfrac{1}{3} +\dfrac{1}{5}-\dfrac{1}{7} + ...) = 4 \sum_{k=1}^{\infty}\dfrac{(-1)^{k+1}}{2k-1}\]

Implement a function called pi_approx to estimate the value of \(\pi\) using the Leibniz method, where the argument must be an integer \(n\) indicating the number of terms added, that is:

\[\pi \approx 4 \sum_{k=1}^{n}\dfrac{(-1)^{k+1}}{2k-1} \]

def pi_approx(n):
    """
    Approximation of $\pi$ using Leibniz's method.

    Parameters
    ----------
    n : int
        Number of terms added.

    Returns
    -------
    output : float
    """

    pi = #FIXME#  # Initialization
    for k in range(#FIXME#):
        numerator = #FIXME#
        denominator = #FIXME#
        pi +=   # Add numerator/denominator to total sum
    return 4 * pi

  Cell In[1], line 15
    pi = #FIXME#  # Initialization
         ^
SyntaxError: invalid syntax

pi_approx(100)

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Cell In[2], line 1
----> 1 pi_approx(100)

NameError: name 'pi_approx' is not defined

Check your answer! pi_approx(100) must return 3.1315929035585537

Exercise 2

Let \(\sigma(n)\) be defined as the sum of the proper divisors of \(n\) (that is, less than \(n\)). Amicable numbers are positive integers \(n_1\) and \(n_2\) such that the sum of the proper divisors of one is equal to the other number and vice versa, that is :

\[\sigma(n_1)=n_2 \quad \textrm{and} \quad \sigma(n_2)=n_1\]

For example, the numbers 220 and 284 are friendly numbers.

• Proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, hence \(\sigma(220) = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284\).

• Proper divisors of 284 are 1, 2, 4, 71 and 142, hence \(\sigma(284) = 1 + 2 + 4 + 71 + 142 = 220\).

Implement the following functions:

• proper_divisors(n) such that returns a list of the proper divisors of \(n\).

• sigma(n) such that it returns the value \(\sigma(n)\).

• friends(n, m) such that it returns True if \(n\) and \(m\) are friendly numbers and False otherwise.

def proper_divisors(n):
    proper_divisors = []
    for i in range(#FIXME#)
        if #FIXME#:  # Hint: Use modulus operation
            proper_divisors.#FIXME#
    return #FIXME#

  Cell In[3], line 3
    for i in range(#FIXME#)
                  ^
SyntaxError: '(' was never closed

proper_divisors(220)

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Cell In[4], line 1
----> 1 proper_divisors(220)

NameError: name 'proper_divisors' is not defined

def sigma(n):
    proper_divisors_list = #FIXME#
    sum = 0
    for x in proper_divisors:
        sum#FIXME#
    return sum

  Cell In[5], line 2
    proper_divisors_list = #FIXME#
                           ^
SyntaxError: invalid syntax

sigma(220)

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Cell In[6], line 1
----> 1 sigma(220)

NameError: name 'sigma' is not defined

def amicables(n, m):
    sigma_n = #FIXME#
    sigma_m = #FIXME#
    if #FIXME#:
        return True
    else:
        return #FIXME#

  Cell In[7], line 2
    sigma_n = #FIXME#
              ^
SyntaxError: invalid syntax

n, m = 220, 284
print(f"Proper divisors of {n} are {proper_divisors(n)}, then sigma({n}) = {sigma(n)}")
print(f"Proper divisors of {m} are {proper_divisors(m)}, then sigma({m}) = {sigma(m)}")
print(f"Are 220 and 284 amicable numbers?: {amicables(n, m)}")

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Cell In[8], line 2
      1 n, m = 220, 284
----> 2 print(f"Proper divisors of {n} are {proper_divisors(n)}, then sigma({n}) = {sigma(n)}")
      3 print(f"Proper divisors of {m} are {proper_divisors(m)}, then sigma({m}) = {sigma(m)}")
      4 print(f"Are 220 and 284 amicable numbers?: {amicables(n, m)}")

NameError: name 'proper_divisors' is not defined