What is a Collatz graph in Python?

A Collatz graph, often visualized using tools like Python, represents the trajectory of numbers under the Collatz conjecture. The Collatz conjecture posits that for any positive integer `n`, repeatedly applying the following rules will eventually lead to `1`:

1. If `n` is even, divide it by 2 (`n = n / 2`).

2. If `n` is odd, multiply it by 3 and add 1 (`n = 3n + 1`).

The Collatz graph visualizes these sequences as a directed graph. Each node represents a number, and an edge from node `a` to node `b` exists if applying the Collatz function to `a` results in `b`.

Here's a simple example in Python to generate part of a Collatz sequence and, conceptually, how it relates to the graph:

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

# Example Usage:
start_number = 6
collatz_sequence = collatz(start_number)
print(collatz_sequence) # Output: [6, 3, 10, 5, 16, 8, 4, 2, 1]


In a graph representation, `6` would have an edge to `3`, `3` would have an edge to `10`, and so on. The full Collatz graph would illustrate how all numbers eventually connect back to `1`, if the conjecture holds true.

Creating a full visualization of the Collatz graph can be complex, particularly for large numbers, but libraries like `NetworkX` can be utilized for graph representation and visualization. Visualizing this graph often involves focusing on subgraphs or specific ranges of numbers to make the relationships clearer. The visualization helps in understanding the interconnectedness of numbers under the Collatz rules.

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