Collatz sequence in Python

The Collatz sequence (also known as the 3n+1 problem) is a mathematical sequence where we repeatedly apply rules to a positive integer until it reaches 1. For any positive integer n:

• If n is even: n = n/2
• If n is odd: n = 3n + 1
• Continue until n = 1

For example, if n = 13, the sequence is: [13, 40, 20, 10, 5, 16, 8, 4, 2, 1] with a length of 10.

Algorithm Steps

To find the length of a Collatz sequence ?

• If num is 0, return 0
• Initialize length = 1
• While num is not equal to 1:
  • Apply the Collatz rule: num = num/2 if even, else 3*num + 1
  • Increment length by 1
• Return the total length

Implementation

class Solution:
    def solve(self, num):
        if num == 0:
            return 0
        length = 1
        while num != 1:
            num = (num // 2) if num % 2 == 0 else (3 * num + 1)
            length += 1
        return length

ob = Solution()
print(ob.solve(13))

10

Step-by-Step Example

Let's trace through the sequence for n = 13 ?

def collatz_sequence(n):
    sequence = [n]
    while n != 1:
        if n % 2 == 0:
            n = n // 2
        else:
            n = 3 * n + 1
        sequence.append(n)
    return sequence

# Generate and display the sequence
result = collatz_sequence(13)
print(f"Sequence: {result}")
print(f"Length: {len(result)}")

Sequence: [13, 40, 20, 10, 5, 16, 8, 4, 2, 1]
Length: 10

Alternative Implementation

Here's a more concise function-based approach ?

def collatz_length(n):
    if n == 0:
        return 0
    
    length = 1
    while n != 1:
        n = n // 2 if n % 2 == 0 else 3 * n + 1
        length += 1
    return length

# Test with different values
test_values = [1, 5, 13, 27]
for val in test_values:
    print(f"Collatz length of {val}: {collatz_length(val)}")

Collatz length of 1: 1
Collatz length of 5: 6
Collatz length of 13: 10
Collatz length of 27: 112

Key Points

• Use integer division (//) instead of float division (/) for even numbers
• The sequence always reaches 1 for positive integers (unproven conjecture)
• Length includes the starting number and ending 1
• Time complexity varies greatly depending on the input value

Conclusion

The Collatz sequence is implemented by repeatedly applying the 3n+1 rule until reaching 1. Use integer division for even numbers and count each step to find the sequence length.