Juggler sequence: start with n. If even: next = ⌊√n⌋. If odd: next = ⌊n^(3/2)⌋. Example: 3→5→11→36→6→2→1. Conjectured to always reach 1, analogous to Collatz but with square roots. The peak can be enormous — starting at 37, the peak exceeds 10^40!
How to Use
Enter starting value:
- Sequence: All terms to 1
- Peak: Maximum value reached
- Length: Steps to reach 1
The Juggler Conjecture
Like Collatz, it's conjectured that every starting value eventually reaches 1. Verified for all starting values up to 10^6. Unlike Collatz, the odd step (n^(3/2)) grows much faster, but the even step (√n) shrinks much faster too.
Peak Records
Starting at 37: peak reaches a number with over 40 digits! Starting at 113: peak > 10^44. The peaks grow extremely erratically. Some small starting values produce enormous peaks before collapsing back to 1.
Step-by-Step Instructions
- 1Enter starting number.
- 2Generate sequence.
- 3Find peak.
- 4Count steps.
- 5Compare to Collatz.