A371671 Convergence speed of n at height 2 (i.e., A369826(n) - A371074(n)).

0, 0, 0, 0, 1, 1, 3, 2, 2, 1, 1, 9, 1, 1, 1, 1, 8, 1, 1, 0, 1, 19, 1, 0, 1, 2, 7, 5, 1, 1, 1, 29, 1, 2, 1, 1, 4, 1, 1, 0, 1, 39, 1, 0, 2, 1, 5, 2, 1, 1, 2, 49, 2, 1, 1, 1, 6, 1, 3, 0, 1, 59, 1, 0, 1, 1, 17, 2, 1, 3, 1, 69, 1, 1, 1, 2, 3, 4, 1, 0, 1, 79, 1, 0

Offset

-1,7

Comments

A sufficient but not necessary condition for having a constant value of the convergence speed of a tetration base n that is not a multiple of 10 (see A317905) is that the height of the hyperexponent is greater than or equal to tilde(v(a))+2, where tilde(v(a)) := v_5(a-1) iff a == 1 (mod 5), v_5(a^2+1) iff a == {2, 3} (mod 5), v_5(a+1) iff a == 4 (mod 5), v_2(a^2-1)-1 iff a == 5 (mod 10), where v_2(x) = A007814(x) and v_5(x) = A112765(x) are the 2-adic and 5-adic valuations, respectively (see "Number of stable digits of any integer tetration", p. 447, Definition 2.1, in Links).

Links

Table of n, a(n) for n=-1..82.

Marco Ripà, Congruence speed of tetration bases ending with 0, arXiv:2402.07929 [math.NT], 2024.

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.

Jorge Jiménez Urroz and José Luis Andrés Yebra, On the Equation a^x == x (mod b^n), Journal of Integer Sequences, Article 09.8.8, 2009.

Wikipedia, Tetration.

Formula

a(n) = A369826(n) - A371074(n).

Example

For n = 5, a(n) = 3 since A369826(n) - A371074(n) = 4 - 1.

Crossrefs

Cf. A002488, A317905 (constant convergence speed), A369826 (n^^2 and n^^3), A371074 (convergence speed at height 1), A370211 (convergence speed at height 3).

Cf. A007814, A112765.

Sequence in context: A238402 A016558 A154395 * A214609 A152159 A341941

Adjacent sequences: A371668 A371669 A371670 * A371672 A371673 A371674

Keyword

nonn,base

Author

Marco Ripà, Apr 02 2024

Status

approved