Gabriele Di Pietro - cp's OEIS frontend

User: Gabriele Di Pietro

Gabriele Di Pietro has authored 2 sequences.

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

5, 7, 43, 101, 107, 149, 151, 157, 193, 199, 251, 257, 293, 307, 349, 401, 443, 449, 457, 499, 557, 593, 599, 601, 607, 643, 701, 743, 751, 757, 857, 907, 1049, 1051, 1093, 1151, 1193, 1201, 1249, 1301, 1307, 1399, 1451, 1493, 1499, 1543, 1549, 1601, 1607, 1657

Offset: 1

Gabriele Di Pietro and Marco Ripà, Apr 17 2025

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.
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.
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.

			a(1) = 5 since 5 is the smallest prime number with a constant congruence speed of at least 2.

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

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.

Also 5 together with A172469.
Union of {5}, A141927, A141932, A141941, A141946.
Cf. A317905, A321131, A373387, A382862.

from sympy import isprime
valid_mod_50 = {1, 7, 43, 49}
result = [5]
n = 6
while len(result) < 1000:
    if isprime(n) and n % 50 in valid_mod_50:
        result.append(n)
    n += 1
print(result)

a(1) = 5. For n >= 2, a(n) = A172469(n-1).

A382862 Prime numbers whose congruence speed of tetration equals 1.

Original entry on oeis.org

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

Marco Ripà and Gabriele Di Pietro, Apr 13 2025

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.

			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.

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

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.

Cf. A000040, A317905, A321131, A373387.

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.

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User: Gabriele Di Pietro

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

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