A slow-growing sequence defined by an unusual recurrence
Journal of integer sequences, Tome 10 (2007) no. 1
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.
@article{JIS_2007__10_1_a4,
author = {van de Bult, Fokko J. and Gijswijt, Dion C. and Linderman, John P. and Sloane, N.J.A. and Wilks, Allan R.},
title = {A slow-growing sequence defined by an unusual recurrence},
journal = {Journal of integer sequences},
year = {2007},
volume = {10},
number = {1},
zbl = {1137.11010},
language = {en},
url = {http://geodesic.mathdoc.fr/item/JIS_2007__10_1_a4/}
}
TY - JOUR AU - van de Bult, Fokko J. AU - Gijswijt, Dion C. AU - Linderman, John P. AU - Sloane, N.J.A. AU - Wilks, Allan R. TI - A slow-growing sequence defined by an unusual recurrence JO - Journal of integer sequences PY - 2007 VL - 10 IS - 1 UR - http://geodesic.mathdoc.fr/item/JIS_2007__10_1_a4/ LA - en ID - JIS_2007__10_1_a4 ER -
%0 Journal Article %A van de Bult, Fokko J. %A Gijswijt, Dion C. %A Linderman, John P. %A Sloane, N.J.A. %A Wilks, Allan R. %T A slow-growing sequence defined by an unusual recurrence %J Journal of integer sequences %D 2007 %V 10 %N 1 %U http://geodesic.mathdoc.fr/item/JIS_2007__10_1_a4/ %G en %F JIS_2007__10_1_a4
van de Bult, Fokko J.; Gijswijt, Dion C.; Linderman, John P.; Sloane, N.J.A.; Wilks, Allan R. A slow-growing sequence defined by an unusual recurrence. Journal of integer sequences, Tome 10 (2007) no. 1. http://geodesic.mathdoc.fr/item/JIS_2007__10_1_a4/