A370775 -id:A370775 - cp's OEIS frontend

A317905 Convergence speed of m^^m, where m = A067251(n) and n >= 2. a(n) = f(m, m) - f(m, m - 1), where f(x, y) corresponds to the maximum value of k, such that x^^y == x^^(y + 1) (mod 10^k).

0, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 6, 1, 1, 3, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 5, 1, 1

Offset: 2

Marco Ripà, Aug 10 2018

It is possible to anticipate the convergence speed of m^^m, where ^^ indicates tetration or hyper-4 (e.g., 3^^4 = 3^(3^(3^3))), simply looking at the congruence (mod 25) of m. In fact, assuming m > 2, a(n) = 1 for any m == 2, 3, 4, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 21, 22, 23 (mod 25), and a(n) >= 2 otherwise.
It follows that 32/45 = 71.11% of the a(n) assume unitary value.
You can also obtain an arbitrary high convergence speed, such as taking the beautiful base b = 999...99 (9_9_9... n times), which gives a(n) = len(b), for any len(b) > 1. Thus, 99...9^^m == 99...9^^(m + 1) (mod m*10^len(b)), as proved by Ripà in "La strana coda della serie n^n^...^n", pages 25-26. In fact, m = 99...9 == 24 (mod 25) and a(m=24) > 1.
From Marco Ripà, Dec 19 2021: (Start)
Knowing the "constant congruence speed" of a given base (a.k.a. the convergence speed of the base m, assuming m > 2) is very useful in order to calculate the exact number of stable digits of all its tetrations of height b > 1. As an example, let us consider all the a(n) such that n is congruent to 4 (mod 9) (i.e., all the tetration bases belonging to the congruence class 5 (mod 10)). Then, the exact number of stable digits (#S(m, b)) of any tetration m^^b (i.e., the number of its last "frozen" digits) such that m is congruent to 5 (mod 10), for any b >= 3, can automatically be calculated by simply knowing that (under the stated constraint) the congruence speed of the m corresponds to the 2-adic valuation of (m^2 - 1) minus 1. Thus, let k = 1, 2, 3, ..., and we have that
If m = 20*k - 5, then #S(m, b > 2) = b*(v_2(m^2 - 1) - 1) + 1 = b*(v_2(m + 1) + 1);
If m = 20*k + 5, then #S(m, b > 2) = (b + 1)*(v_2(m^2 - 1) - 1) = (b + 1)*(v_2(m - 1));
If m = 5, then #S(m, 1) = 1, #S(m, 2) = 4, #S(m, b > 2) = 8 + 2*(b - 3).
(End)
For any n > 2, the value of a(n) depends on the congruence modulo 18 of n, since the constant congruence speed of m arises from the 14 nontrivial solutions of the fundamental equation y^5 = y in the (commutative) ring of decadic integers (e.g., y = -1 = ...9999 is a solution of y^5 = y, so it originates the law a(n) = min(v_2(m + 1), v_5(m + 1)) concerning every n belonging to the congruence class 0 modulo 18, as stated in the "Formula" section of the present sequence). - Marco Ripà, Feb 17 2022
a(n) satisfies the following multiplicative constraint: for each pair (m_1, m_2) of terms of A067251, a(m_1*m_2) is necessarily greater than or equal to the minimum between a(m_1) and a(m_2) (see Equation 2.4 and Appendix of "A Compact Notation for Peculiar Properties Characterizing Integer Tetration" in Links). - Gabriele Di Pietro, Apr 29 2025

			For m = 25, a(23) = 3 implies that 25^^(25 + i) freezes 3*i "new" rightmost digits (i >= 0).

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

Michel Marcus, reducetower.gp (from Math StackExchange).
Math StackExchange, Calculating a^n (mod m) in the general case
Marco Ripà, On the Convergence Speed of Tetration, ResearchGate (2018).
Marco Ripà, On the constant congruence speed of tetration, Notes on Number Theory and Discrete Mathematics, Volume 26, 2020, Number 3, Pages 245—260.
Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43-61.
Marco Ripà and Gabriele Di Pietro, A Compact Notation for Peculiar Properties Characterizing Integer Tetration, Zenodo, 2025.
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, A349425, A370211, A370775, A371129, A371671, A371720, A372490, A373387.

Cf. A000007, A018247, A018248, A063006, A091661, A091663, A091664, A120817, A120818, A290372, A290373, A290374, A290375.

\\ uses reducetower.gp from links
f2(x,y) = my(k=0); while(reducetower(x, 10^k, y) == reducetower(x, 10^k, y+1), k++); k;
f1(n) = polcoef(x*(x+1)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1) / ((x-1)^2*(x^2+x+1)*(x^6+x^3+1)) + O(x^(n+1)), n, x); \\ A067251
a(n) =  my(m=f1(n)); f2(m, m) - f2(m, m-1);
lista(nn) = {for (n=2, nn, print1(a(n), ", "););} \\ Michel Marcus, Jan 27 2021

Let n > 2. For any integer c >= 0, if n is an element of the set {5, 7, 14, 17, 22, 23, 24, 29, 32, 39, 41, 45, 46}, then a(n + 45*c) >= 2; whereas a(n) = 1 otherwise. - Marco Ripà, Sep 28 2018
If n == 5 (mod 9), then a(n) = v_2(a(n)^2 - 1) - 1, where v_2(x) indicates the 2-adic valuation of x. - Marco Ripà, Dec 19 2021
If n == 1 (mod 18) and n <> 1, then a(n) = min(v_2(m - 1), v_5(m - 1)) (i.e., 1 plus the number of trailing zeros, if any, next to the rightmost digit of m);
if n == 10 (mod 18), then a(n) = min(v_2(m + 1), v_5(m - 1));
if n == {2,8}(mod 9) and n <> 2, then a(n) = v_5(m^2 + 1);
if n == {3,7}(mod 18), then a(n) = min(v_2(m + 1), v_5(n^2 + 1));
if n == {12,16}(mod 18), then a(n) = min(v_2(m - 1), v_5(n^2 + 1));
if n == 4 (mod 9), then a(n) = v_5(m + 1);
if n == 5 (mod 18), then a(n) = v_2(m - 1);
if n == 14 (mod 18), then a(n) = v_2(m + 1);
if n == 6 (mod 9), then a(n) = v_5(m - 1);
if n == 9 (mod 18), then a(n) = min(v_2(m - 1), v_5(m + 1));
if n == 0 (mod 18), then a(n) = min(v_2(m + 1), v_5(m + 1)) (i.e., number of digits of the rightmost repunit "9's" of m); where v_2(x) and v_5(x) indicates the 2-adic valuation of (x) and the 5-adic valuation of (x), respectively. - Marco Ripà, Feb 17 2022

Edited by Jinyuan Wang, Aug 30 2020

A373387 Constant congruence speed of the tetration base n (in radix-10), or -1 if n is a multiple of 10.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 2, 1, 1, -1, 1, 1, 1, 1, 4, 1, 1, 2, 1, -1, 1, 1, 1, 2, 3, 2, 1, 1, 1, -1, 1, 2, 1, 1, 2, 1, 1, 1, 1, -1, 1, 1, 2, 1, 2, 1, 1, 1, 2, -1, 2, 1, 1, 1, 3, 1, 3, 1, 1, -1, 1, 1, 1, 1, 6, 1, 1, 3, 1, -1, 1, 1, 1, 2, 2, 2, 1, 1, 1, -1, 1, 2, 1

Offset: 0

Marco Ripà, Jun 02 2024

It has been proved that this sequence contains arbitrarily large entries, while a(0) = a(1) = 0 by definition (given the fact that 0^0 = 1 is a reasonable choice and then 0^^b is 1 if b is even, whereas 0^^b is 0 if b is even). For any nonnegative integer n which is not a multiple of 10, a(n) is given by Equation (16) of the paper "Number of stable digits of any integer tetration" (see Links).
Moreover, a sufficient condition for having a constant congruence speed of any tetration base n, greater than 1 and not a multiple of 10, is that b >= 2 + v(n), where v(n) is equal to
  u_5(n - 1)        iff n == 1 (mod 5),
  u_5(n^2 + 1)      iff n == 2,3 (mod 5),
  u_5(n + 1)        iff n == 4 (mod 5),
  u_2(n^2 - 1) - 1  iff n == 5 (mod 10)
  (u_5 and u_2 indicate the 5-adic and the 2-adic valuation of the argument, respectively).
Therefore b >= n + 1 is always a sufficient condition for the constancy of the congruence speed (as long as n > 1 and n <> 0 (mod 10)).
As a trivial application of this property, we note that the constant congruence speed of the tetration 3^^b is 1 for any b > 1, while 3^3 is not congruent to 3 modulo 10. Thus, we can easily calculate the exact number of the rightmost digits of Graham’s number, G(64) (see A133613), that are the same of the homologous rightmost digits of 3^3^3^... since 3^3 is not congruent to 3 modulo 10, while the congruence speed of n = 3 is constant from height 2 (see A372490). This means that the last slog_3(G(64))-1 digits of G(64) are the same slog_3(G(64))-1 final digits of 3^3^3^..., whereas the difference between the slog_3(G(64))-th digit of G(64) and the slog_3(G(64))-th digit of 3^3^3^... is congruent to 6 modulo 10.
The constant congruence speed of tetration satisfies the following multiplicative constraint: for each pair (n_1, n_2) of nonnegative integers whose product is not divisible by 10, a(n_1*n_2) is necessarily greater than or equal to the minimum between a(n_1) and a(n_2) (see Equation 2.4 and Appendix of "A Compact Notation for Peculiar Properties Characterizing Integer Tetration" in Links). - Marco Ripà, Apr 26 2025

			a(3) = 1 since 3^^b := 3^3^3^... freezes 1 more rightmost digit for each unit increment of b, starting from b = 2.

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, Pages 245—260.
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 Gabriele Di Pietro, A Compact Notation for Peculiar Properties Characterizing Integer Tetration, Zenodo, 2025.
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, Graham's Number.
Wikipedia, Tetration.

Cf. A067251, A133613, A317824, A317903, A317905, A349425, A370211, A370775, A371129, A371671, A372490.

Cf. A000007, A018247, A018248, A063006, A091661, A091663, A091664, A120817, A120818, A290372, A290373, A290374, A290375.

def v2(n):
    count = 0
    while n % 2 == 0 and n > 0:
        n //= 2
        count += 1
    return count
def v5(n):
    count = 0
    while n % 5 == 0 and n > 0:
        n //= 5
        count += 1
    return count
def V(a):
    mod_20 = a % 20
    mod_10 = a % 10
    if mod_20 == 1:
        return min(v2(a - 1), v5(a - 1))
    elif mod_20 == 11:
        return min(v2(a + 1), v5(a - 1))
    elif mod_10 in {2, 8}:
        return v5(a ** 2 + 1)
    elif mod_20 in {3, 7}:
        return min(v2(a + 1), v5(a ** 2 + 1))
    elif mod_20 in {13, 17}:
        return min(v2(a - 1), v5(a ** 2 + 1))
    elif mod_10 == 4:
        return v5(a + 1)
    elif mod_20 == 5:
        return v2(a - 1)
    elif mod_20 == 15:
        return v2(a + 1)
    elif mod_10 == 6:
        return v5(a - 1)
    elif mod_20 == 9:
        return min(v2(a - 1), v5(a + 1))
    elif mod_20 == 19:
        return min(v2(a + 1), v5(a + 1))
def generate_sequence():
    sequence = []
    for a in range(1026):
        if a == 0 or a == 1:
            sequence.append(0)
        elif a % 10 == 0:
            sequence.append(-1)
        else:
            sequence.append(V(a))
    return sequence
sequence = generate_sequence()
print("a(0), a(1), a(2), ..., a(1025) =", ", ".join(map(str, sequence)))

a(n) = -1 iff n == 0 (mod 10), a(n) = 0 iff n = 1 or 2. Otherwise, a(n) >= 1 and it is given by Equation (16) from Ripà and Onnis.

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, ...

A379906 Smallest integer greater than 1 and not ending in 0 whose congruence speed is not constant at height n (see A373387).

Original entry on oeis.org

2, 2, 5, 307, 807, 72943, 795807, 1295807, 16295807, 166295807, 16666295807, 31666295807, 81666295807, 8581666295807, 26581907922943, 503581666295807, 2003581666295807, 90476581907922943, 140476581907922943, 6847003581666295807, 61847003581666295807, 911847003581666295807

Offset: 1

Marco Ripà, Jan 05 2025

The present sequence is a subsequence of A068407.
Although the congruence speed of any integer m > 1 not divisible by 10 is certainly stable at height m + 1 (for a tighter upper bound see "Number of stable digits of any integer tetration" in Links), this sequence contains infinitely many terms, implying the existence of infinitely many tetration bases whose congruence speed does not stabilize in less than b + 1 iterations, for any chosen positive integer b.
As a nontrivial example, the congruence speed of m := 45115161423787862411847003581666295807 becomes stable at height 41, which exactly matches the mentioned tight bound, for the numbers ending in 2, 3, 7, or 8, of v_5(45115161423787862411847003581666295807^2 + 1) + 2, where v_5(...) indicates the 5-adic valuation of the argument.

			a(5) = 807 since the congruence speed of 807 is 0 at height 1, 4 at heights 2, 3, 4, and 5, finally matching the value of the constant congruence speed of 807 at height 6 (and it is the smallest integer whose congruence speed stabilizes at height 6 or above).

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, Pages 245—260.
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.
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. A068407, A290372, A290373, A290374, A290375, A317905, A370211, A370775, A371129, A371671, A372490, A373387.

As long as ceiling(log_10(a(n))) < n, for any n > 3, the least significant floor(log_10(a(n))) digits of a(n) (from right to left) are given by the first floor(log_10(a(n))) entries of A290372(n), or A290373(n), or A290374(n), or A290375(n) (i.e., all but the first digit of each a(n) are described by ({5^2^k}_oo + {2^5^k}_oo) := ...17196359523418092077057, ({5^2^k}_oo - {2^5^k}_oo) := ...37588152996418333704193, (- {5^2^k}_oo + {2^5^k}_oo) := ...2411847003581666295807, and (- {5^2^k}_oo - {2^5^k}_oo) := ...2803640476581907922943).

A380031 Smallest integer of d digits, greater than 1 and not ending in 0, whose constant congruence speed is not yet constant at height d + 2.

Original entry on oeis.org

5, 807, 81666295807, 81907922943, 161423787862411847003581666295807, 115161423787862411847003581666295807, 45115161423787862411847003581666295807, 44317662666830362972182803640476581907922943, 776138023544317662666830362972182803640476581907922943

Offset: 1

Marco Ripà, Jan 10 2025

The present sequence is a subsequence of A068407, but it is not a subsequence of A379906 (e.g., a(4) is not a term of A379906).
Although the congruence speed of any integer m > 1 not divisible by 10 is certainly stable at height m + 1 (for a tighter upper bound see "Number of stable digits of any integer tetration" in Links), this sequence contains infinitely many terms, implying the existence of infinitely many tetration bases of d digits whose congruence speed does not stabilize in less than d + 3 iterations (e.g., the congruence speed of 807, a 3-digit number, becomes constant only at height).
As a nontrivial example, the congruence speed of a(10) := 712222747129609220545115161423787862411847003581666295807 (a 57-digit number whose constant congruence speed is also 57) becomes stable at height 60, which exactly matches the mentioned tight bound, for the numbers ending in 2, 3, 7, or 8, of v_5(712222747129609220545115161423787862411847003581666295807^2 + 1) + 2, where v_5(...) indicates the 5-adic valuation of the argument.
The smallest integer of d digits whose constant congruence speed is not yet constant at height d + 3 is 435525708925199660525680385844696084258785712222747129609220545115161423787862411847003581666295807 (a 99-digit number whose congruence speed stabilizes at height 104 to its constant value of 101).
For any n >= 2, terms of this sequence derive from one digit 5 that appears in any of the two 10-adic solutions (- {5^2^k}_oo + {2^5^k}_oo) :=  ...2411847003581666295807 and (- {5^2^k}_oo - {2^5^k}_oo) := ...2803640476581907922943 of the fundamental 10-adic equation y^5 = y (see "The congruence speed formula" in Links). The only other candidate terms can arise from the remaining two symmetric 10-adic solutions ({5^2^k}_oo + {2^5^k}_oo) := ...7196359523418092077057 and ({5^2^k}_oo - {2^5^k}_oo) := ...7588152996418333704193 of y^5 = y as particular patterns of 0s and 5 may occur in the corresponding (neverending) strings (e.g., '50...0').
Consequently, if n > 1 is given, a(n) is always congruent modulo 50 to 7 or 3.

			a(2) = 807 since the corresponding 10-adic solution of y^5 = y is ...61423787862411847003581666295807 where the rightmost digit 5 appears to the left side of a(2) itself, while no smaller numbers with the same feature are achievable by cutting the 10-adic integer ...30362972182803640476581907922943 (also one of the 15 solutions of the fundamental 10-adic equation y^5 = y) in correspondence of its rightmost digit 5.

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, Pages 245—260.
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.
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. A068407, A290372, A290373, A290374, A290375, A317905, A370211, A370775, A371129, A371671, A372490, A373387, A379906.

For any n > 1, a(n) corresponds to a cut on the right side of a digit 5 that appears inside one of the two strings (- {5^2^k}_oo - {2^5^k}_oo) := ...96579486665776138023544317662666830362972182803640476581907922943 and (- {5^2^k}_oo + {2^5^k}_oo) := ...84258785712222747129609220545115161423787862411847003581666295807, or even to a cut on the right side of a 5 belonging to rare digit-patterns consisting of juxtaposed 5 and trailing 0's occurring inside ({5^2^k}_oo + {2^5^k}_oo) := ...7196359523418092077057 or ({5^2^k}_oo - {2^5^k}_oo) := ...7588152996418333704193.

cp's OEIS Frontend

A317905 Convergence speed of m^^m, where m = A067251(n) and n >= 2. a(n) = f(m, m) - f(m, m - 1), where f(x, y) corresponds to the maximum value of k, such that x^^y == x^^(y + 1) (mod 10^k).

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A373387 Constant congruence speed of the tetration base n (in radix-10), or -1 if n is a multiple of 10.

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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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A379906 Smallest integer greater than 1 and not ending in 0 whose congruence speed is not constant at height n (see A373387).

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A380031 Smallest integer of d digits, greater than 1 and not ending in 0, whose constant congruence speed is not yet constant at height d + 2.

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