This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A377124 #25 Dec 27 2024 11:14:02 %S A377124 1,6,1,6,5,6,1,6,1,1,1,6,1,6,5,6,1,6,1,6,1,6,1,6,5,6,1,6,1,1,1,6,1,6, %T A377124 5,6,1,6,1,6,1,6,1,6,5,6,1,6,1,5,1,6,1,6,5,6,1,6,1,6,1,6,1,6,5,6,1,6, %U A377124 1,1,1,6,1,6,5,6,1,6,1,6,1,6,1,6,5,6,1 %N A377124 Phase shift (original name "sfasamento") of the tetration base 10*n at any height greater than or equal to 3. %C A377124 Let m^^b be m^m^...^m b-times (integer tetration). %C A377124 For any n, the phase shift of n*10 at height b is defined as the congruence class modulo 10 of the difference between the least significant non-stable digit of (n*10)^^b and the corresponding digit of (n*10)^^(b+1), so the phase shift of n*10 at height 1 is trivially A065881(n) while the phase shift of n*10 at height 2 is given by A376838(n). %C A377124 Thus, assume b >= 3 and, for any given tetration base n*10, this sequence represents the congruence classes modulo 10 of the differences between the rightmost non-stable digit of (n*10)^^b and the zero of (n*10)^^(b+1) which occupies the same decimal position (counting from right to the left) as the rightmost nonzero digit of (n*10)^^b (see Appendix of "Graham's number stable digits: an exact solution" in Links). %C A377124 If n == 3,7 (mod 10), a(n) <> A065881(n) since the least significant nonzero digit of (n*10)^^b only depends on the last digit of n^^(b - 1) and, in the mentioned two cases, n*10 is not congruent to 0 modulo 4, whereas (n*10)^(n*10) is clearly a multiple of 4 given the fact that it is also a multiple of 100 (e.g., if n = 3 is given, the last nonzero digit of (n*10)^(n*10) is 3 iff (n*10) == 1 (mod 4), 9 iff (n*10) == 2 (mod 4), 7 iff (n*10) == 3 (mod 4), 1 iff (n*10) == 0 (mod 4), which is the only case we are considering here since (3*10)^(3*10) == 0 (mod 100)). %D A377124 Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011. ISBN 978-88-6178-789-6. %H A377124 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 A377124 Marco Ripà, <a href="https://arxiv.org/abs/2402.07929">Congruence speed of tetration bases ending with 0</a>, arXiv:2402.07929 [math.NT], 2024. %H A377124 Marco Ripà, <a href="https://arxiv.org/abs/2411.00015">Graham's number stable digits: an exact solution</a>, arXiv:2411.00015 [math.GM], 2024. %H A377124 Marco Ripà, <a href="https://www.researchgate.net/publication/387314761_Twelve_Python_Programs_to_Help_Readers_Test_Peculiar_Properties_of_Integer_Tetration">Twelve Python Programs to Help Readers Test Peculiar Properties of Integer Tetration</a>, ResearchGate, 2024. See pp. 18, 19, 20, 27. %H A377124 Wikipedia, <a href="https://en.wikipedia.org/wiki/Graham's_number">Graham's Number</a>. %H A377124 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>. %F A377124 a(n) equals the least significant nonzero digit of n^((n*10)^(n*10)). %F A377124 Let h indicate the least significant nonzero digit of n. Then, %F A377124 a(n) = 1 iff h = 1,3,7,9; %F A377124 a(n) = 5 iff h = 5; %F A377124 a(n) = 6 iff h = 2,4,6,8. %e A377124 a(1) = 1 since 10^(10^10) == 0 (mod 10^10000000000) and 10^(10^10) == 1 (mod 10^10000000001), and trivially 1 - 0 = 1. %Y A377124 Cf. A065881, A317905, A370211, A371671, A373387, A376838. %K A377124 nonn,base,easy %O A377124 1,2 %A A377124 _Marco Ripà_, Oct 17 2024