The Modified Recursive Sum Sequence Problem (MRSSP):

Steps:

Define a sequence 𝑆 (𝑛) as follows:

1. Start with any positive integer 𝑛.

2. If 𝑛 is even, let the next number in the sequence be the sum of all digits of 𝑛.

3. If 𝑛 is odd, let the next number in the sequence be 3𝑛+1

4. Repeat the process indefinitely.

Explanation:

In the Modified Recursive Sum Sequence Problem (MRSSP), the behavior of the sequence can vary significantly depending on the starting integer. Unlike the Collatz Conjecture, which suggests that any positive integer will eventually reach 1, the MRSSP does not have a proven universal behavior.

For example, starting with ( n = 14 ), the sequence eventually enters a loop: ( 14 -> 5 -> 16 -> 7 -> 22 -> 4 -> 4. Here, the sequence does not reach 1 but instead stabilizes at 4.

The MRSSP can produce sequences that either:

- Enter a repeating cycle (like the example above).
- Potentially grow indefinitely if the sequence keeps generating larger numbers.

Due to the nature of the rules, some sequences might never reach 1, especially if they fall into a repeating cycle or continue to grow.

The MRSSP Conjecture is a cool problem, (even though it uses the same equation as the collatz, it has a different set of rules,) that can't really have a definitive answer, try it for yourself!