Path records

The Collatz conjecture asserts that a sequence defined by repeatedly applying the function $$ T(n) = \begin{cases} (3n+1)/2 &\text{ if $n$ is odd, or}\\ n/2 &\text{ if $n$ is even} \end{cases} $$ will always converge to the cycle passing through the number 1 for arbitrary initial positive integer $n$. Define $t(n)$ the highest number occurring in the sequence starting at $n$. The $n$ is called the path record if for all $m < n$ the inequality $t(m) < t(n)$ holds. The following table lists the known path records for all $n$ below $1536 \times 2^{60}$. $B(n)$ and $B(t(n))$ are the sizes of their arguments measured in the number of bits. Lagarias and Weiss (1992) predicted using the theory for random walks that $$ \limsup\limits_{n\rightarrow\infty} \frac{\log t(n)}{\log n} = 2 \text{.}$$ In other words, the highest number occurring in the sequence for a path record $n$ grows like $n^2$. This agrees with the empirical evidence given in the table below.

References

• Lagarias, J. C.; Weiss, A. The $3x+1$ Problem: Two Stochastic Models. Ann. Appl. Probab. 2 (1992), no. 1, 229–261. doi:10.1214/aoap/1177005779.
• Sequence A006884 in The On-Line Encyclopedia of Integer Sequences.
• Eric Roosendaal, 3x + 1 Path Records.

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