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 A386008 #12 Jul 20 2025 16:48:57 %S A386008 -2,-1,-1,2,1,-2,-1,0,9,1,-1,0,-1,1,1,0,-4,0,19,1,-2,-1,-2,3,2,-1,-1, %T A386008 0,29,1,-2,0,-1,1,1,0,-2,0,39,1,-2,-2,-1,2,1,-1,-1,0,49,1,-1,0,-1,1,1, %U A386008 0,-2,0,59,1,-2,-1,-1,6,1,-1,-3,0,69,1,-1,0,-2,1,2 %N A386008 a(n) = A386005(n) - n*A373387(n). %C A386008 Assuming that n is not a multiple of 10, this sequence measures the difference between the number of stable digits in n^^n and the product of n times the constant congruence speed of n (see A373387 and A317905). %C A386008 A negative value of a(n) means that, on average, each iteration n^^b --> n^^(b+1), with b < A372490(n), contributes fewer stable digits than what will be contributed per step once the congruence speed of n reaches its constant value. %C A386008 Positive values also imply the existence of a pre-period for the given n, and indicate that its average contribution per step exceeds the constant congruence speed of n. %C A386008 If n == 5 (mod 10) and n <> 5, the pre-period of the congruence speed always has length 2 (i.e., A372490(n) = 3). However, the number of stable digits observed up to that point follows two distinct rules: if n = 20*k + 5 (for positive integer k), then a(n) = (n + 1)*A373387(n); if n = 20*(k - 1) + 15, then it is n*A373387(n) + 1. The resulting residue is A373387(n) in the former case, and 1 in the latter. For n = 5, the pre-period has length 3 (and this is the only such case for n ending in 5). %D A386008 Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011. ISBN 978-88-6178-789-6. %H A386008 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 A386008 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 A386008 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a> %e A386008 a(3) = -1 since 3^3^3 == 3^3^3^3 (mod 10^2) while 3^3^3 <> 3^3^3^3 (mod 10^3), and the constant congruence speed of 3 is equal to 1. Thus, a(3) = 2 - 3*1. %Y A386008 Cf. A317905, A349425, A356946, A372490, A373387, A386005. %K A386008 sign,base,hard %O A386008 2,1 %A A386008 _Marco Ripà_, Jul 14 2025