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A381253 Prime numbers whose constant congruence speed of tetration is greater than 1.

%I A381253 #24 Apr 29 2025 13:28:35
%S A381253 5,7,43,101,107,149,151,157,193,199,251,257,293,307,349,401,443,449,
%T A381253 457,499,557,593,599,601,607,643,701,743,751,757,857,907,1049,1051,
%U A381253 1093,1151,1193,1201,1249,1301,1307,1399,1451,1493,1499,1543,1549,1601,1607,1657
%N A381253 Prime numbers whose constant congruence speed of tetration is greater than 1.
%C A381253 The only positive integers with a constant congruence speed greater than 1 (see A373387) are necessarily congruent to 1, 7, 43, or 49 modulo 50.
%C A381253 As a result, 36% of positive integers have a constant congruence speed of at least 2, while 20% of primes have a constant congruence speed greater than 1. In the interval (1, 10^4), there are 1229 prime numbers, 247 of whom have a constant congruence speed of at least 2.
%C A381253 Moreover, as a consequence of Dirichlet's theorem on arithmetic progressions, Theorem 3 of "The congruence speed formula" (see Links) proves that, for any given positive integer k, there are infinitely many primes characterized by a constant congruence speed of (exactly) k.
%D A381253 Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011. ISBN 978-88-6178-789-6
%H A381253 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 A381253 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 A381253 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 A381253 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>.
%F A381253 a(1) = 5. For n >= 2, a(n) = A172469(n-1).
%e A381253 a(1) = 5 since 5 is the smallest prime number with a constant congruence speed of at least 2.
%o A381253 (Python)
%o A381253 from sympy import isprime
%o A381253 valid_mod_50 = {1, 7, 43, 49}
%o A381253 result = [5]
%o A381253 n = 6
%o A381253 while len(result) < 1000:
%o A381253     if isprime(n) and n % 50 in valid_mod_50:
%o A381253         result.append(n)
%o A381253     n += 1
%o A381253 print(result)
%Y A381253 Also 5 together with A172469.
%Y A381253 Union of {5}, A141927, A141932, A141941, A141946.
%Y A381253 Cf. A317905, A321131, A373387, A382862.
%K A381253 nonn,base
%O A381253 1,1
%A A381253 _Gabriele Di Pietro_ and _Marco Ripà_, Apr 17 2025