A337836 - OEIS

%I #73 Feb 08 2022 08:04:28

%S 8,18,68,2318,7318,1068,32318,501068,7532318,3626068,23157318,

%T 120813568,3538782318,1097376068,110960657318,49925501068,

%U 1880980188568,355101282318,53760863001068,15613890344818,587818480188568,2495167113001068

%N 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.

%C 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)).

%C From _Marco Ripà_, Dec 31 2021: (Start)

%C 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

%C 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).

%C (End)

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

%H Marco Ripà, <a href="https://doi.org/10.7546/nntdm.2020.26.3.245-260">On the constant congruence speed of tetration</a>, Notes on Number Theory and Discrete Mathematics, 2020, 26(3), 245-260.

%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, <a href="https://www.researchgate.net/publication/357447814_Number_of_stable_digits_of_any_integer_tetration">Number of stable digits of any tetration</a>, ResearchGate, December 2021.

%F 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)].

%F a(n) = 5-adic valuation of a(n)^2 + 1. - _Marco Ripà_, Dec 31 2021

%e 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).

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

%K base,nonn

%O 1,1

%A _Marco Ripà_, Sep 24 2020