A321131 -id:A321131 - cp's OEIS frontend

A321130 Values of m (mod 25) such that V(m) >= 2, where V(m) indicates the constant convergence speed of the tetration base m.

0, 1, 5, 7, 15, 18, 24

Offset: 1

Marco Ripà, Oct 27 2018

This sequence represents the values of the base a such that a^^m, where ^^ indicates tetration or hyper-4 (e.g., 3^^4=3^(3^(3^3))), is characterized by a convergence speed at or above 2 (fast m-adic convergence). Only 26% of the positive integers belong to this list (see A317905).

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

Marco Ripà, On the constant congruence speed of tetration, Notes on Number Theory and Discrete Mathematics, Volume 26, 2020, Number 3, pp. 245—260.
Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43-61.
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. A067251, A317824, A317903, A317905, A321131.

For m = 57, m (mod 25) == 7 and 7^^n has a convergence speed greater than 1, since A317905(m = 57) = 3 > 1 and also A317905(m = 7) = 2 > 1.

A370775 Integers m whose (constant) convergence speed is exactly 2 (i.e., m^^(m+1) has 2 more rightmost frozen digits than m^^m, where ^^ indicates tetration).

Original entry on oeis.org

5, 7, 18, 24, 26, 32, 35, 43, 45, 49, 51, 74, 75, 76, 82, 85, 93, 99, 107, 115, 118, 125, 132, 143, 149, 151, 155, 157, 165, 168, 174, 176, 195, 199, 201, 205, 207, 218, 224, 226, 232, 235, 243, 245, 251, 257, 268, 274, 275, 276, 282, 285, 293, 299, 301, 307

Offset: 1

Marco Ripà, May 01 2024

It is well known (see Links) that as the hyperexponent of the integer m becomes sufficiently large, the constant convergence speed of m is the number of new stable digits that appear at the end of the result for any further unit increment of the hyperexponent itself, and a sufficient (but not necessary) condition to get this fixed value is to set the hyperexponent equal to m plus 1.

			If n = 2, m = 7 and so 7^^8 has exactly 2 more stable digits at the end of the result than 7^^7.

Marco Ripà, On the constant congruence speed of tetration, Notes on Number Theory and Discrete Mathematics, Volume 26, 2020, Number 3, Pages 245—260 (see Table 1, pp. 249—251).
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 (see Equation 16, p. 454).
Wikipedia, Tetration

Cf. A317905 (convergence speed of m^^m), A321130, A321131, A371129.

a(n) is such that A317905(m) = 2, for m = 5, 6, 7, ...

A371129 Integers m whose (constant) convergence speed is exactly 3 (i.e., m^^(m+1) has 3 more rightmost frozen digits than m^^m, where ^^ indicates tetration).

Original entry on oeis.org

25, 55, 57, 68, 105, 124, 126, 135, 185, 193, 215, 249, 265, 295, 318, 345, 374, 375, 376, 425, 432, 455, 505, 535, 568, 585, 615, 665, 682, 695, 745, 751, 775, 807, 818, 825, 855, 874, 876, 905, 932, 935, 943, 985, 999, 1001, 1015, 1057, 1065, 1095, 1124

Offset: 1

Marco Ripà, May 01 2024

It is well known (see Links) that as the hyperexponent of the integer m becomes sufficiently large, the constant convergence speed of m is the number of new stable digits that appear at the end of the result for any further unit increment of the hyperexponent itself, and a sufficient (but not necessary) condition to get this fixed value is to set the hyperexponent equal to m plus 1 (e.g., if n := 3, m = 57 and so 57^^58 has exactly 3 more stable digits at the end of the result than 57^^57).

			If n = 3, m = 57 and so 57^^58 has exactly 3 more stable digits at the end of the result than 57^^57.

Marco Ripà, On the constant congruence speed of tetration, Notes on Number Theory and Discrete Mathematics, Volume 26, 2020, Number 3, Pages 245—260 (see Table 1, pp. 249—251).
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 (see Equation 16, p. 454).
Wikipedia, Tetration

Cf. A317905 (convergence speed of m^^m), A321130, A321131, A370775.

a(n) is such that A317905(m) = 3, for m = 25, 26, 27, ...

A354959 Tetration bases with a constant convergence speed >= 3.

Original entry on oeis.org

15, 25, 55, 57, 65, 68, 95, 105, 124, 126, 135, 145, 175, 182, 185, 193, 215, 225, 249, 255, 265, 295, 305, 318, 335, 345, 374, 375, 376, 385, 415, 425, 432, 455, 465, 495, 505, 535, 545, 568, 575, 585, 615, 624, 625, 626, 655, 665, 682, 695, 705, 735, 745

Offset: 1

Marco Ripà, Jul 23 2022

The convergence speed of any integer greater than 1 and not divisible by 10 is constant if and only if we are considering an integer tetration and its constant convergence speed is greater than 2 if and only if the tetration base is of the form m + k*1000, for k >= 0, where m is a term.

			57 is a term since the constant convergence speed of 57 is 3 and (trivially) 57 has no trailing zeros.

Marco Ripà, Table of n, a(n) for n = 1..10000
Marco Ripà, On the constant congruence speed of tetration, Notes on Number Theory and Discrete Mathematics, Volume 26, 2020, Number 3, pp. 245—260.
Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43-61.
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. A317905, A337392, A337833, A321130, A321131.

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.

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

Original entry on oeis.org

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).

cp's OEIS Frontend

A321130 Values of m (mod 25) such that V(m) >= 2, where V(m) indicates the constant convergence speed of the tetration base m.

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A370775 Integers m whose (constant) convergence speed is exactly 2 (i.e., m^^(m+1) has 2 more rightmost frozen digits than m^^m, where ^^ indicates tetration).

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A371129 Integers m whose (constant) convergence speed is exactly 3 (i.e., m^^(m+1) has 3 more rightmost frozen digits than m^^m, where ^^ indicates tetration).

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A354959 Tetration bases with a constant convergence speed >= 3.

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A382862 Prime numbers whose congruence speed of tetration equals 1.

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

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