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 A339313 #31 Dec 16 2021 04:12:14
%S A339313 2,5,193,1249,22943,2218751,4218751,74218751,574218751,30000000001,
%T A339313 281907922943,581907922943,6581907922943,123418092077057,
%U A339313 480163574218751,19523418092077057,40476581907922943,2152996418333704193,23640476581907922943,3640476581907922943
%N A339313 Smallest prime numbers characterized by a convergence speed of n, assuming a(1) = 2 (since 2^2 <> 2^2^2 (mod 10) and 2^2^2 == 2^2^2^2 (mod 10)).
%C A339313 It is possible to prove that for any integer n >= 1 there are infinitely many prime numbers with a convergence speed equal to n (invoking Dirichlet's theorem on arithmetic progressions and considering the bases of the form 10^j - 1 + (2*k)*10^j = (2*k + 1)*10^j - 1, since their convergence speed is always equal to j and 10 never divides (2*k + 1)).
%C A339313 Since the only base with a convergence speed of 0 is a = 1 (and 1 is not a prime number), this sequence starts from a(1) = 2, while the convergence speed of 2 has been assumed to be 1 because the tetration 2^^b "freezes" one more rightmost digit for any unitary increment of b for any b >= 3 (the "constant" convergence speed of 2 is 1, even if V(2) = 0 according to the definition used in A317905). In general, a sufficient but not necessary condition to find the constant convergence speed of the base a, is to assume b >= a + 1 (e.g., V(2) corresponds to the new rightmost frozen digit going from 2^^(b >= 3) to 2^^(b + 1)).
%C A339313 This is not a strictly increasing sequence, since 3640476581907922943 = a(20) < a(19) = 23640476581907922943 (while a(19) < a(21) = 803640476581907922943).
%C A339313 For any n >= 3, a(n) == {1,3,7,9}(mod 10), since any prime above 5 is coprime to 10.
%H A339313 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 A339313 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.
%e A339313 For n = 3, a(3) = 193, since 193 is the smallest prime number which is characterized by a convergence speed of 3.
%Y A339313 Cf. A000040, A063006, A091661, A290372, A290373, A290374, A290375, A317905, A337833.
%K A339313 nonn,hard
%O A339313 1,1
%A A339313 _Marco Ripà_, Nov 29 2020