A340345 a(n) is the smallest base of the form 2 + 10*k which is characterized by a convergence speed of n, where A317905(n) represents the convergence speed of m^^m.

2, 32, 432, 182, 5182, 30182, 123932, 1061432, 280182, 15905182, 74498932, 367467682, 1344030182, 23316686432, 11109655182, 255250280182, 1170777623932, 7274293248932, 22533082311432, 175120972936432, 365855836217682, 7041576051061432

Offset: 1

Author

Marco Ripà, Jan 04 2021

Keywords

• base
• nonn

Comments

Let n >= 1. For any t == 2 (mod 10), if 5^n divides (t^2 + 1) and 5^(n + 1) does not divide (t^2 + 1), then V(t) = n (where V(t) represents the convergence speed of t). In particular, the aforementioned property holds for any a(n), since a(n) belongs to the residue class 2 modulo 10 for any n. Moreover, 5^n always divides (a(n) + A337836(n)).

From Marco Ripà, Dec 31 2021: (Start)

In general, any tetration base m = A067251(n) which is congruent to {2,8}(mod 10) is characterized by a convergence speed equal to the 5-adic valuation of m^2 + 1. Similarly, if m is congruent to 4(mod 10), then the convergence speed of m is given by m + 1, whereas if m belongs to the congruence class 6 modulo 10, then its convergence speed is m - 1. Lastly, for any m congruent to 5 modulo 10, the congruence speed exceeds by 1 the 2-adic valuation of m^2 - 1.

Moreover, assuming m > 1, m^m is not congruent to m^m^m if and only if m belongs to the congruence class 2 modulo 20 or 18 modulo 20, whereas if m = A067251(n) is not coprime to 10 and is not equal to 5, then the number of new stable digits from m^m^m to m^m^m^m is always equal to the convergence speed of m. The aforementioned statement, in general, is untrue if m is coprime to 10 (see "Number of stable digits of any integer tetration" in the Links section).

(End)

Examples

			For n = 4, a(4) = 182 is characterized by a convergence speed of 4, and it is the smallest base such that V(a) = 4. Moreover, 5 has to divide a(4)^2+1 exactly four times (i.e., a(4)^2+1 = 33125 = 5^4*53 is a multiple of 5^4 and is not divisible by 5^5).
		

References

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

Links

• Marco Ripà, On the constant congruence speed of tetration, Notes on Number Theory and Discrete Mathematics, 2020, 26(3), 245-260.
• Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43-61.
• Marco Ripà, Number of stable digits of any tetration, ResearchGate, December 2021.

Crossrefs

Cf. A317905, A337392, A337833, A337836, A349425.

Formula

a(n) = g(n) + u(n), where g(n) = (2^5^n (mod 10^n)) (mod 2*5^n) and where u(n) = [0 iff g(n) <> g(n + 1); 2*5^n iff g(n) = g(n + 1)].

a(n) = 5-adic valuation of a(n)^2 + 1. - Marco Ripà, Dec 31 2021