A390535 - OEIS

%I #60 Jan 24 2026 23:13:48

%S 2,2,2,6,2,2,7,3,2,2,7,2,2,3,9,2,3,2,2,23,2,3,30,2,2,34,2,2,38,3,2,42,

%T 3,2,23,3,2,2,2,3,2,3,41,2,3,2,31,2,2,2,9,2,3,7,2,2,74,2,3,2,39,3,2,

%U 41,3,2,2,86,2,3,2,3,71,2,3,2,2,2,102,2,3,106

%N Smallest integer > 1 whose radix-A013929(n) congruence speed never stabilizes.

%C 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).

%C Furthermore, for every integer n >= 1, a(n) exists (and, generally, 2 <= a(n) <= A013929(n)/2 holds) since a(n) cannot exceed the product of the distinct prime factors of A013929(n) (which, by definition, only consists of nonsquarefree terms). Indeed, in the radix-A013929(n) numeral system, the tetration a(n)^^k, for k = 2,3,4,... will eventually be congruent to 0 modulo A013929(n), and hence the radix-A013929(n) congruence speed of a(n) will rapidly accelerate for each unit increment of k.

%C If we restrict attention to integers a(n) that are coprime to A013929(n), no explicit upper bounds are currently known for the terms of the analogous sequence "Smallest integer > 1 and coprime to A013929(n) whose radix-A013929(n) congruence speed never stabilizes".

%D Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011. ISBN 978-88-6178-789-6.

%H Gabriele Di Pietro, <a href="/A390535/b390535.txt">Table of n, a(n) for n = 1..194</a>

%H Marco Ripà, <a href="https://doi.org/10.7546/nntdm.2021.27.4.43-61">The congruence speed formula</a>, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43—61.

%H Marco Ripà and Gabriele Di Pietro, <a href="https://doi.org/10.5281/zenodo.17626007">A Compact Notation for Peculiar Properties Characterizing Integer Tetration</a>, Zenodo, 2025.

%H Marco Ripà and Luca Onnis, <a href="https://doi.org/10.7546/nntdm.2022.28.3.441-457">Number of stable digits of any integer tetration</a>, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441—457.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>.

%F For any given n > 1, let A013929(n) = p_1^q_1*p_2^q_2*...*p_k^q_k be the prime factorization of A013929(n), where p_1, p_2, ..., p_k are distinct primes. Then, a(n) <= p_1*p_2*...*p_k.

%e a(4) = 6 since the congruence speed of 6 does not converge to a fixed value in the radix-12 numeral system (it equals 0 at height 1, 3 at height 2, 23325 at height 3, and so forth).

%Y Cf. A013929, A317905, A373387 (radix-10 constant congruence speed), A379906, A380031, A390597, A390598 (radix-6 constant congruence speed).

%K nonn,hard

%O 1,1

%A _Marco Ripà_ and _Gabriele Di Pietro_, Dec 07 2025