A337836 a(n) is the smallest base of the form 8 + 10*k which is characterized by a convergence speed of n, where A317905(n) represents the convergence speed of m^^m.

8, 18, 68, 2318, 7318, 1068, 32318, 501068, 7532318, 3626068, 23157318, 120813568, 3538782318, 1097376068, 110960657318, 49925501068, 1880980188568, 355101282318, 53760863001068, 15613890344818, 587818480188568, 2495167113001068

Offset

1,1

Comments

Let n >= 1. For any t == 8 (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 8 modulo 10 for any n. Moreover, 5^n always divides (a(n) + A340345(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 18 modulo 20 or 2 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)

References

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

Links

Table of n, a(n) for n=1..22.

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.

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

Example

For n = 3, a(3) = 68 is characterized by a convergence speed of 3, and it is the smallest base such that V(a) = 3. Moreover, 5^3 has to divide a(3) (i.e., a(3)^2+1 = 4625 = 5^3*37 is a multiple of 5^3).

Crossrefs

Cf. A317905, A337392, A337833, A340345, A349425.

Sequence in context: A341528 A032795 A120543 * A036747 A151351 A354769

Adjacent sequences: A337833 A337834 A337835 * A337837 A337838 A337839

Keyword

base,nonn

Author

Marco Ripà, Sep 24 2020

Status

approved