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A379906 Smallest integer greater than 1 and not ending in 0 whose congruence speed is not constant at height n (see A373387).

%I A379906 #18 Jan 18 2025 09:26:44
%S A379906 2,2,5,307,807,72943,795807,1295807,16295807,166295807,16666295807,
%T A379906 31666295807,81666295807,8581666295807,26581907922943,503581666295807,
%U A379906 2003581666295807,90476581907922943,140476581907922943,6847003581666295807,61847003581666295807,911847003581666295807
%N A379906 Smallest integer greater than 1 and not ending in 0 whose congruence speed is not constant at height n (see A373387).
%C A379906 The present sequence is a subsequence of A068407.
%C A379906 Although the congruence speed of any integer m > 1 not divisible by 10 is certainly stable at height m + 1 (for a tighter upper bound see "Number of stable digits of any integer tetration" in Links), this sequence contains infinitely many terms, implying the existence of infinitely many tetration bases whose congruence speed does not stabilize in less than b + 1 iterations, for any chosen positive integer b.
%C A379906 As a nontrivial example, the congruence speed of m := 45115161423787862411847003581666295807 becomes stable at height 41, which exactly matches the mentioned tight bound, for the numbers ending in 2, 3, 7, or 8, of v_5(45115161423787862411847003581666295807^2 + 1) + 2, where v_5(...) indicates the 5-adic valuation of the argument.
%D A379906 Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011. ISBN 978-88-6178-789-6.
%H A379906 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, Volume 26, 2020, Number 3, Pages 245—260.
%H A379906 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 A379906 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.
%H A379906 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 A379906 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>.
%F A379906 As long as ceiling(log_10(a(n))) < n, for any n > 3, the least significant floor(log_10(a(n))) digits of a(n) (from right to left) are given by the first floor(log_10(a(n))) entries of A290372(n), or A290373(n), or A290374(n), or A290375(n) (i.e., all but the first digit of each a(n) are described by ({5^2^k}_oo + {2^5^k}_oo) := ...17196359523418092077057, ({5^2^k}_oo - {2^5^k}_oo) := ...37588152996418333704193, (- {5^2^k}_oo + {2^5^k}_oo) :=  ...2411847003581666295807, and (- {5^2^k}_oo - {2^5^k}_oo) := ...2803640476581907922943).
%e A379906 a(5) = 807 since the congruence speed of 807 is 0 at height 1, 4 at heights 2, 3, 4, and 5, finally matching the value of the constant congruence speed of 807 at height 6 (and it is the smallest integer whose congruence speed stabilizes at height 6 or above).
%Y A379906 Cf. A068407, A290372, A290373, A290374, A290375, A317905, A370211, A370775, A371129, A371671, A372490, A373387.
%K A379906 nonn,base,hard
%O A379906 1,1
%A A379906 _Marco Ripà_, Jan 05 2025