A382862 Prime numbers whose congruence speed of tetration equals 1.

2, 3, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 103, 109, 113, 127, 131, 137, 139, 163, 167, 173, 179, 181, 191, 197, 211, 223, 227, 229, 233, 239, 241, 263, 269, 271, 277, 281, 283, 311, 313, 317, 331, 337, 347, 353, 359

Offset: 1

Author

Marco Ripà and Gabriele Di Pietro, Apr 13 2025

Keywords

• nonn
• base

Comments

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.

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.

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.

Examples

			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.
		

References

• Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011. ISBN 978-88-6178-789-6.

Links

• Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43—61.
• Marco Ripà, Twelve Python Programs to Help Readers Test Peculiar Properties of Integer Tetration, ResearchGate, 2024. See pp. 22-23, 27.
• Marco Ripà and Luca Onnis, Number of stable digits of any integer tetration, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441—457.
• Wikipedia, Tetration.

Crossrefs

Cf. A000040, A317905, A321131, A373387.

Formula

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).

Terms of A000040 congruent modulo 25 to one term of A321131.