OFFSET
1,1
COMMENTS
For a fixed integer n > 1, the radix-n congruence speed of every integer m > 1 not a multiple of n stabilizes (to a positive integer constant) if and only if n is squarefree (see A373387 for the radix-10 definition and A390598 for the constant congruence speed in radix-6).
For every n, a(n) >= A390535(n) holds by definition.
If we restrict attention to integers a(n) that are coprime to A013929(n), no explicit upper bounds are currently known.
REFERENCES
Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011. ISBN 978-88-6178-789-6.
LINKS
Gabriele Di Pietro, Table of n, a(n) for n = 1..194
Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43—61.
Marco Ripà and Gabriele Di Pietro, A Compact Notation for Peculiar Properties Characterizing Integer Tetration, Zenodo, 2025.
Marco Ripà and Luca Onnis, Number of stable digits of any integer tetration, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441—457.
Wikipedia, Tetration.
EXAMPLE
a(1)=7 since the congruence speed of 7 does not converge to a fixed value in the radix-4 and 7 is coprime to 4.
CROSSREFS
KEYWORD
nonn,hard
AUTHOR
Gabriele Di Pietro and Marco Ripà, Dec 09 2025
STATUS
approved
