cp's OEIS Frontend

A356946 Number of stable digits of the integer tetration n^^n (i.e., maximum nonnegative integer m such that n^^n is congruent modulo 10^m to n^^(n + 1)).

%I A356946 #12 Sep 11 2022 19:26:38
%S A356946 1,0,2,3,12,7,12,7,9
%N A356946 Number of stable digits of the integer tetration n^^n (i.e., maximum nonnegative integer m such that n^^n is congruent modulo 10^m to n^^(n + 1)).
%C A356946 a(10) = 10^^9 is too large to include. In general, if n is a multiple of 10, then a(n) is given by the number of trailing zeros that appear at the end of n^^n.
%C A356946 This follows from the constancy of the "congruence speed" (AKA "convergence speed" here on the OEIS) of hyper-3 for any exponentiation base which is a multiple of 10, otherwise the congruence speed is constant only for hyper-4 and it is strictly positive for any tetration base n >= 1 that is not a multiple of 10 (for an explicit formula to calculate a(n) for any n, see the linked paper entitled "Number of stable digits of any integer tetration").
%D A356946 Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011. ISBN 978-88-6178-789-6.
%H A356946 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 A356946 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 A356946 Marco Ripà and Luca Onnis, <a href="https://doi.org/10.7546/nntdm.2022.28.3.441-457">Number of stable digits of any integer tetration</a>, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441-457.
%H A356946 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>
%e A356946 For n = 3, 3^3^3 is congruent to 3^3^3^3 (mod 10^2) and 3^3^3 is not congruent to 3^3^3^3 (mod 10^3). Thus, a(3) = 2.
%Y A356946 Cf. A317824, A317903, A317905, A349425.
%K A356946 nonn,base
%O A356946 1,3
%A A356946 _Marco Ripà_, Sep 05 2022