A390535 Smallest integer > 1 whose radix-A013929(n) congruence speed never stabilizes.

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

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).

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.

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".

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.

Formula

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.

Example

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).

Crossrefs

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

Sequence in context: A361301 A240501 A278251 * A286847 A291439 A023957

Adjacent sequences: A390532 A390533 A390534 * A390536 A390537 A390538

Keyword

nonn,hard

Author

Marco Ripà and Gabriele Di Pietro, Dec 07 2025

Status

approved