Which Collatz numbers do Busy Beavers simulate (if any)?

Fibra · October 16, 2025, 6:49pm

The following is a comparison between the tape development generated by n = 371581 (left - using Collatz’s Tape; to be described in a bit) and \mathbf{BB(4)} (right - using the Turing Machine: 1RB1LB_1LA0LC_1RH1LD_1RD0RA):

The tape on the left is generated in the following manner:

You consider an empty tape with all unmarked cells, such that the reading head (standing initially in the middle of the tape) applies the collatz function to a starting n:

f(n) = \begin{cases} n/2 & \text{if} \quad n \equiv 0 \quad (\text{mod}\, 2) \\ (3n + 1)/2 & \text{if} \quad n \equiv 1 \quad (\text{mod}\, 2) \\ \end{cases}

flipping the state of the cell it currently stands at (at position x) in the following manner:

S(x) = \begin{cases} 0 & \text{if} \quad n \equiv 0 \quad (\text{mod}\, 3) \\ 1 - S(x) & \text{if} \quad n \equiv 1 \quad (\text{mod}\, 3) \\ 1 & \text{if} \quad n \equiv 2 \quad (\text{mod}\, 3) \\ \end{cases}

Additionally, the reading head moves left if n is odd, and right if n is even. It will do this until n = 1 is reached. The tape development can then be seen over time (↓). E.g. n = 10^{20}:

Albeit, there are vast differences between the examples shown initially, this comparison was made because it’s known that many Busy Beaver champions (and candidates) display Collatz-like behaviour.

However, do (should) Busy Beavers themselves simulate any Collatz number(s) (using the version of Collatz’s Tape displayed before)?

(Un)related interesting Collatz tapes

10^{20} + 2

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10^{20} - 1