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A382862 Prime numbers whose congruence speed of tetration equals 1.

%I A382862 #38 Apr 24 2025 13:33:09
%S A382862 2,3,11,13,17,19,23,29,31,37,41,47,53,59,61,67,71,73,79,83,89,97,103,
%T A382862 109,113,127,131,137,139,163,167,173,179,181,191,197,211,223,227,229,
%U A382862 233,239,241,263,269,271,277,281,283,311,313,317,331,337,347,353,359
%N A382862 Prime numbers whose congruence speed of tetration equals 1.
%C A382862 The only positive integers with a constant congruence speed of 1 (see A373387) are necessarily congruent to 2, 3, 4, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 21, 22, or 23 modulo 25.
%C A382862 Thus, a prime number is characterized by a unit constant congruence speed if and only if it is not congruent to 1, 7, 43, or 49 modulo 50.
%C A382862 As a result, (16*4)% of positive integers have a constant congruence speed of 1, while (16*5)% of primes have a unit constant congruence speed (since the mentioned constraint excludes all the multiples of 5). In the interval (1, 10^4) there are 1229 prime numbers, 982 of whom have a unit constant congruence speed.
%D A382862 Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011. ISBN 978-88-6178-789-6.
%H A382862 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 A382862 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. 22-23, 27.
%H A382862 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 A382862 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>.
%F A382862 a(1) = 2, a(2) = 3. For any n >= 3, a(n) : A000040(m) == 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97, 103, 109, 113, 119, 121, 127, 131, 133, 137, 139 (mod 150).
%F A382862 Terms of A000040 congruent modulo 25 to one term of A321131.
%e A382862 a(3) = 11 since the 2 and 3 have a unit constant congruence speed, while the constant congruence speed of 5 and 7 equals 2.
%Y A382862 Cf. A000040, A317905, A321131, A373387.
%K A382862 nonn,base
%O A382862 1,1
%A A382862 _Marco Ripà_ and _Gabriele Di Pietro_, Apr 13 2025