Python Day 3: Finding Fun Numbers

All the maths fans know the story about ** 1729**, the

**Ramanujan**Number.

One day Hardy visited Ramanujan in the hospital and said he took a taxi with a boring licence number, 1729. Ramanujan said: it is a very

**interesting number**; it is the

*smallest number expressible as the sum of two cubes in two different ways*.

**1729 = 1 ^{3} + 12^{3} = 9^{3} + 10^{3}**

We will find some interesting number, the **sum of divisors’ square number**, and **perfect number** with *python*.

**Perfect Number**:

a **perfect number** is a positive integer that is equal to the sum of its proper positive divisors

**Example**: the proper divisor of 6 is 1,2,3 and **6 = 1+2+3**.

**Python code:**

#we want to find all the perfect number within 1000. sieve= [1]*(1000+1) n = 2 while n <= 1000: # check n if sieve[n] == n: print(n, "is a perfect number") # add n to all k * n where k > 1 kn = 2 * n while kn <= 1000: sieve[kn] += n kn += n n += 1

**Result:**

We can see **6,28** and **496** are perfect numbers.

**The sum of divisors’ square number:**

Suppose we want to find some number such that the **Sum of the squares of the number’s divisors is a perfect square number**

**Example:**

The divisors of *42* are *1,2,3,6,7,14,21,42*

the square of divisors are *1,4,9,36,49, 196, 441, 1764*

Sum of the square of divisors are *1+4+9+ 36+49+196+441+1764=2500*

2500=500 * 500 is a** perfect square**

**Python Code:**

def list_squared(m,n): list=[] for i in range(m,n+1): sum=0 s_list=[] for j in range(1,int(i**0.5)+1): if i%j==0: div=i/j sum+=j**2 if j!=div: sum+=div**2 sqt=sum**0.5 if int(sqt)==sqt: s_list=[i,sum] list.append(s_list) return list list_squared(42,250)

**Output:**

We can see **42**, and **246** satisfy the conditions in the range (42,250)

I like Number theory because I feel it entitled every number a meaning 🙂 In recent years, my friends and I got a little sensitive and sentimental about our ages. However, when i look at the numbers, every year is beautiful indeed:

27, the **Cube** age,

28, the **Perfect** age,

29, a **Gaussian** age,

30, a smallest **2,3,5,** divisor age,

31, a **PRIME** age,

32, a **Binary** age!!!

I like my number theory classmate JOHNES said:

Appreciated every little things in life, like

Tuesdays, it only comes once a week!

**Happy Studying! 🐼**