## Python: Collatz Sequence

**Python: Longest Collatz Sequence **

**Problem No.14 in ProjectEuler**

**Definition** Wikipedia: Start with any positive integer n. Then each term is obtained from the previous term as follows: if the previous term is even, the next term is one half the previous term. If the previous term is odd, the next term is 3 times the previous term plus 1. The conjecture is that no matter what value of n, the sequence will always reach 1.

So the formula is: Starting with n, the next n will be:

n/2 (if n is even)

3n + 1 (if n is odd)

If we start with 13, we generate the following sequence:

13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1.

**A walk through: **

n=13, 13 is odd, then n = 13 * 3 + 1 , n= 40

n=40, 40 is even, then n= 40/2, n=20

n=20, 20 is even, then n=20/2, n=10

n=10, 10 is even, then n=10/2, n=5

n=5, 5 is odd, then n=5*3+1 , n=16

n=16, 16 is even, then n=16/2, n=8

n=8, 8 is even, then n=8/2, n=4

n=4, 4 is even, then n=4/2, n=2

n=2, 2 is even, then n=2/2, n=1

n=1 then end of sequence

**The Task:** The task in ProjectEuler is to searching for a the Number N, under one million, that produces the longest chain.

Overview to my python cases:In my company, we are not allowed to download any software, so i don’t have any Python platform. To solve this i am using an online python interpreter, some time it’s become slow. So in this code (and others) i am spiting the range in 10 each with 100,000 then running the code to get the longest chain in each range. So the Number N, under one million, that produces the longest chain is:

**The Answer: **In my previous codes or math solving challenges in pybites or ProjectEuler I am solving the problems, writing the code, but not posting my answer to ProjectEuler platform. Today, and with Problem No.14 i decide to post the answer in the ProjectEuler platform for the first time just to see what will happen. The answer was 837799, and I get this page.

*In the code bellow, i set the range from 1 to 50000.*

**The Code:**

chain2=[]

longest=[0,0]

def collatz_Seq(num):

t= num

chain=[num]

while t !=1 :

if t%2==0:

t=t/2

chain.append(int(t))

else:

t=3*t+1

chain.append(int(t))

return chain

for num in range (1,50000):

chain2 = collatz_Seq(num)

if len(chain2) > longest[0]:

longest[0] = len(chain2)

longest[1] = num

chain2=[]

print(‘num:’,longest[1],’ has a longest chain: ‘,longest[0])

## Python: The Factorial

**Python: Factors of the Number N**

This is a short task to get the factors of a given number. The Definition of Factors of N is: The pairs of numbers you multiply to get the N number.

For instance, factors of 15 are 3 and 5, because 3×5 = 15. Some numbers have more than one factorization (more than one way of being factored). For instance, 12 can be factored as 1×12, 2×6, or 3×4

In this task we will write a Python code to ask the user for a number N then will get all the pairs number that if we multiply them will get that N number, we will store the pairs in a array ‘factors’.

**The Code:**

def factors_of_n(num):

a=1

factors=[]

while a <= num:

if num%a==0:

if (num/a,a) not in factors:

factors.append((a,int(num/a)))

a = a + 1

return factors

#Ask the user for a number

num=int(input(“Enter a number: “))

print(factors_of_n(num))

## Python: Amicable Numbers

**Python: Amicable Numbers **

** Problem No.21 on projectEuler **

In this task we need to calculate the SUM of all divisors of N, from 1 to N.

Then we calculate the Sum of (sum of N divisors ) let’s say M .

Now if

**For example**, the proper divisors of **A=220 **are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110; therefore F(220) = 284. And the proper divisors of **B=284 **are 1, 2, 4, 71 and 142; so F(284) = 220. So F(A) =B and F(B)=A then A and B are amicable.

Definition:If f(a) = b and f(b) = a, where a ≠ b,

thena and b are anamicable pairand each of a and b are calledamicable numbers.

**In this task** we will ask the user to enter a Range of numbers and we will search for all Amicable Numbers pairs in that range(from – to) if we fond one we will print out the number and the divisors list. The main function here is the one that get the sum of divisors, we will call it get_divisors_sum and we will examine Amicable with If statement.

Hint ..

To check if the sum of divisors from both said are equal we use (t2==x )

AND that this sum are not same we use (t1!=t2)

AND that we did not print the pair before we use: (t2 not in ami_pair)

**The Code:**

#Python: Amicable Numbers

#Problem No.21 on projectEuler

num1=int(input(‘The range Start from:’))

num2=int(input(‘The range Ends at:’))

t1=0

t2=0

l=[]

l1=[]

l2=[]

ami_pair=[]

def get_divisors_sum (num):

t=0

l=[]

for a in range (1,num):

if num%a==0 :

l.append(a)

t=t+a

return t,l

for x in range(num1,num2):

l1=[]

t1,l1=get_divisors_sum(x)

l2=[]

t2,l2=get_divisors_sum(t1)

if (t2==x) and (t1!=t2) and (t2 not in ami_pair):

print(‘\nget_divisors_sum({}) is {} divisors={}’.format(x,t1,l1))

print(‘\nget_divisors_sum({}) is {} divisors={}’.format(t1,t2,l2))

print(‘\nSo {} and {} are Amicable Numbers .’.format(t1,t2))

ami_pair.extend((t1,t2))

## Python Project: Sum of power of digits

**Python: Sum of power of digits **

** Problem No.16 **

In Problem No.16, projectEuler ask to find the Sum of power of digits in the number, for example if we have 2^15 (2 to power of 15) the answer is 23768 then we need to calculate the sum of this number (2+3+7+6+8 ) that’s equal to 26.

**The Task:** So our task in this project is to find the sum of the digits of (2^1000). To write this as a program and to make it more __general__ we will ask the user to input the number and the power he want, then I start thinking to restrict user from input large numbers that could cause CPU problems, but then i decide to keep it open as is.

**The Code:**

num=2

p=15

num=int(input(“Enter a number “))

p=int(input(“Enter a power “))

a=num**p

def sum_of_digits(num,p):

a=num**p

l=[int(i) for i in str(a)]

print(l)

for x in range (len(l)):

t=t+l[x]

print (“sum =”,t)

## Python: Prime Numbers in Range

**Python: Prime Numbers in a Range**

Our task here is simple as the title, we will have a range and will test each number to see if it is a prime then we will add it to a list, once we finish we will print out the list.

To complete this task we will use one of our function we create last time (**Read:** is prime post).

So, here we will ask the user to input two numbers num1 and num2 the we will pass all the numbers in the range to ** is_prime()** and store the result in a list.

**The Code:**

#Function to get all Prime numbers in a range.

#Ask the user to enter two numbers

print ‘Get all Prime numbers in a range\n’

num1=int(input(“Enter the first number in the range: “))

num2=int(input(“Enter the last number in the range: “))

#create the list

prime_list=[]

def get_all_prime(num1,num2):

for x in range (num1,num2):

if is_prime(x) ==’Prime’:

prime_list.append(x)

return prime_list

#Call the function and print the list

get_all_prime(num1,num2)

print prime_list

print len(prime_list)

## Python Project

**Python: Sum of the Square and Square of the Sum **

**Difference of sum of the square and the square of the sum**

I fond this on projecteuler.net Usually I add some steps to there problems to make it more application look and feel. Later on, we’ll see how.

**Problem assumption**: If we said that we have a range of numbers (1,10) then the sum of this range is 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55, Now the Square of the sum is 55^2; thats mean the Square of the sum of (55) = 3025.

And for the same range, the Sum of Square means that we will get the Square of each number in the range then will get there summation. So with our range (1,10) Sum of Square is 1^2 + 2^2 + 3^2 + 4^2 + 5^2 …. + 10^2 = 385

The Problem #6 in ProjectEulerThe sum of the squares of the first ten natural numbers is,1

^{2}+ 2^{2}+ … + 10^{2}= 385The square of the sum of the first ten natural numbers is,(1 + 2 + … + 10)

^{2}= 55^{2}= 3025

Hencethedifference betweenthe sum of the squares of the first ten natural numbers and the square of the sum is 3025 − 385 = 2640.

The Task:Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.

**Inhancment** Now to make this task workking as application and to get more general output of it, first we will ask the user to input a range of the numbers, then we will applay the function on that range.

**The Code:**

# Problem 6 in ProjectEuler.

def square_of_sum(num1, num2):

tot = 0

for x in range(num1, num2+1):

tot = tot+x

print(‘The Square of the Sun’,tot*tot)

return tot*tot

def sum_of_square(num1, num2):

tot = 0

for x in range(num1, num2+1):

tot = tot+(x*x)

print(‘The Sum of the Square ‘, tot)

return tot

#Ask the user for his input.

num1 = int(input(‘Enter First number in the range: ‘))

num2 = int(input(‘Enter the last number in the range: ‘))

#Call the functions.

w1 = square_of_sum(num1, num2)

w2 = sum_of_square(num1, num2)

#Output the Difference.

print(‘The Difference is: ‘, w1-w2)

## Python Project

**Python: Prime Numbera **

** Is it Prime**

In a simple way, a **Prime** number is a number that cannot be made by multiplying other numbers. So 4 is **Not Prime **because we can say that 4 = 2 x 2, 6 is **Not Prime** because it can be produced by multiplying 3 x 2; but 5 **Is Prime** because we can’t find any integer numbers that can produce 5.

Numbers such as 1, 3, 5, 7, 11, 13 … all are Prime Numbers.

**T**his function will ask the user to write a number then we will examine it to see whether it is a prime or not.

**The Code:**

num=int(input(“Enter a number: “))

def is_prime(num):

result=”Prime”

for t in range (2,num):

if num%t==0 :

result=”Not Prime”

break

return result

print num,’is’,is_prime(num)