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A Slow-Growing Sequence Defined by an Unusual Recurrence

Fokko J. van de Bult·Dion C. Gijswijt·John P. Linderman·N. J. A. Sloane·Allan R. Wilks
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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.

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Fokko J. van de Bult, Dion C. Gijswijt, John P. Linderman, N. J. A. Sloane, Allan R. Wilks. 2006-02-22. A Slow-Growing Sequence Defined by an Unusual Recurrence. https://arxiv.org/abs/math/0602498

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