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, … 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, …) appears for the first time at about position 2 ↑ (2 ↑ (3 ↑ (4 ↑ (5 ↑ … ↑ ((t − 2) ↑ (t − 1)))))), where ↑ denotes exponentiation. The final section discusses generalizations.