Abstract:

The sequence starts with $a(1) =1$; to extend it one writes the sequence so far as $XY^k$, where $X$ and $Y$ are strings of integers, $Y$ is nonempty and $k$ is as large as possible: then the next term is $k$. The sequence begins 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, $\ldots$ A $4$ appears for the first time at position 220, but a $5$ does not appear until about position $10^{10^{23}}$. The main result of the paper is a proof that the sequence is unbounded. We also present results from extensive numerical investigations of the sequence and of certain derived sequences, culminating with a heuristic argument that $t$ (for $t=5,6, \ldots$) appears for the first time at about position $2\uparrow (2\uparrow (3\uparrow (4\uparrow (5\uparrow \ldots \uparrow ({(t-2)}\uparrow {(t-1)})))))$, where $\uparrow$ denotes exponentiation. The final section discusses generalizations.