https://file.aiquickdraw.com/mathbot/file/1728923686633rifp65fm.jpg

The function shown in the image is a definition of the Collatz function, commonly associated with the Collatz conjecture. It operates as follows:

\[
f_c(n) = 
\begin{cases} 
3n + 1 & \text{if } n \text{ is odd} \\ 
\frac{n}{2} & \text{if } n \text{ is even} \\ 
\text{stop} & \text{if } n = 1 
\end{cases}
\]

### Explanation
1. **If \( n \) is odd**: The function applies \( 3n + 1 \).
2. **If \( n \) is even**: The function applies \( \frac{n}{2} \).
3. **If \( n = 1 \)**: The function stops.

This function is used to illustrate the Collatz sequence. Starting from any positive integer \( n \), applying \( f_c \) repeatedly is hypothesized to eventually reach the number 1, regardless of the starting integer. This has been checked for very large values of \( n \), but a proof or disproof that it holds for all positive integers remains unknown.

Would you like a detailed example of how this function works on a specific number, or further exploration of the Collatz conjecture?

### Related Questions
1. How can the Collatz sequence be visualized for a given number?
2. What are the longest known Collatz sequences for small integers?
3. Is there any known proof that the Collatz sequence always reaches 1 for all integers?
4. What are the practical applications of the Collatz conjecture?
5. How does this sequence behave for large values of \( n \)?

### Tip
Try applying this function to different starting values to observe how quickly they reach 1. This is a fun way to experiment with the Collatz conjecture!

The query submitted by the user concerns a function defined by the Collatz conjecture, which applies the following rules based on whether a number is odd, even, or equal to 1.

Learn how the Collatz function works by defining recursive steps based on whether a number is odd, even, or equal to 1. This function illustrates the Collatz conjecture, a famous unsolved problem in mathematics. Suitable for high school students exploring recursion and number theory.

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