A371048 -id:A371048 - cp's OEIS frontend

A349425 n-th stable digit (in decimal system) of n^(n^(...^n)).

1, 3, 3, 8, 0, 2, 5, 9, 3, 0, 7, 4, 5, 8, 6, 7, 8, 3, 1, 0, 1, 4, 4, 8, 7, 8, 0, 6, 4, 0, 9, 8, 0, 1, 0, 3, 5, 8, 7, 0, 6, 8, 2, 4, 1, 0, 3, 3, 3, 0, 2, 3, 5, 2, 3, 6, 8, 7, 0, 0, 7, 3, 3, 5, 0, 8, 0, 0, 8, 0, 7, 8, 8, 0, 3, 0, 6, 9, 1, 0, 7, 7, 6, 6, 5, 7, 3

Offset: 1

Marco Ripà, Nov 17 2021

The integer tetration (or hyper-4) n^^b is characterized by a well-known property involving its rightmost digits (as b grows an increasing number of the rightmost digits of n^^b are frozen - following the general rule described by Equation (16) of the linked paper "Number of stable digits of any integer tetration", p. 454).
In 2011 Ripà conjectured that, for any n >= 1, if b >= n + 2, then the n rightmost digits of n^^b are stable.
The above-mentioned paper, published in 2022, proved that this conjecture is true and also stated the stronger sufficient condition that the height of the hyperexponent is greater than or equal to tilde(v(a)) + 2, where tilde(v(a)) := v_5(a - 1) iff a == 1 (mod 5), v_5(a^2 + 1) iff a == {2, 3} (mod 5), v_5(a + 1) iff a == 4 (mod 5), v_2(a^2 - 1) - 1 iff a == 5 (mod 10), where v_2(x) = A007814(x) and v_5(x) = A112765(x) are the 2-adic and 5-adic valuations of x, respectively. - Marco Ripà, Jul 24 2024

			For n = 3, a(3) = 3 since 3^^5 == 387(mod 10^3). Thus, (387(mod 10^3) - 387(mod 10^2))/10^2 = 3.

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, 2020, 26(3), 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. A007814, A112765, A120962, A317824, A317903, A317905, A369624, A371048, A371078, A373387.

b:= proc(n) option remember; local m, v, w; m, w:= 10^n, n;
      do v:= n&^w mod m; if w=v then return v else w:=v fi od
    end:
a:= n-> `if`(irem(n, 10)=0, 0, iquo(b(n), 10^(n-1))):
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 17 2021

def A349425(n):
    if n % 10 == 0: return 0
    m, n1, n2 = n, 10**n, 10**(n-1)
    while (k := pow(n,m,n1)) != m: m = k
    return k//n2 # Chai Wah Wu, Dec 19 2021

a(n) = (n^^(n + 2)(mod 10^n) - n^^(n + 2)(mod 10^(n - 1)))/10^(n - 1).

A372490 Minimum integer such that the convergence speed of the tetration n^^(a(n)) is constant (i.e., the convergence speed of n^^(a(n) + m) is a fixed number for any positive integer m) and -1 if n is a multiple of 10.

Original entry on oeis.org

3, 2, 2, 4, 3, 2, 2, 1, -1, 2, 2, 1, 2, 3, 2, 1, 3, 1, -1, 2, 3, 2, 2, 3, 3, 2, 2, 1, -1, 2, 2, 1, 2, 3, 2, 1, 3, 1, -1, 2, 3, 2, 2, 3, 3, 2, 2, 1, -1, 2, 2, 1, 2, 3, 2, 1, 3, 1, -1, 2, 3, 2, 2, 3, 3, 2, 2, 1, -1, 2, 2, 1, 2, 3, 3, 1, 3, 1, -1, 2, 3, 2, 2, 3

Offset: 2

Marco Ripà, May 03 2024

The common digits taken into account by the convergence speed formula might consider leading 0's (such as at n = 51 or n = 57) and they are included in the total (see comments of A371048 and A371074).
Although the present sequence has some recursive features, this is not a periodic sequence.
Furthermore, for each positive integer k there exists n such that a(n) = k. E.g., if k := 6, then it is sufficient to take n = 807 since a(807) = 6 (see Links). In detail, a(807) = 6 since 807^^1 = 807, 807^^2 == 8331827388850173[7943] (mod 10^20), 807^^3 == 668229405256[3285]7943 (mod 10^20), 807^^4 == 74874632[6260]32857943 (mod 10^20), 807^^5 == 4897[3150]626032857943 (mod 10^20), and finally a(807) = 6 follows from 807^^6 ==1[228]3150626032857943 (mod 10^20) and 807^^7 == 62283150626032857943 (mod 10^20), given the fact that v_5(807^2 + 1) + 2 = 4 + 2 = 6 is a sufficient condition on the hyperexponent.

			For n := 2, a(n) = 3 since 2^^1 = 2, 2^^2 = 4, and finally 2^^3 = 2^(2^2) = 16 is congruent modulo 10 to 65536 = 2^^4 (while 2^^3 is not congruent modulo 10^2 to 2^^4) so that the congruence speed of 2^^b is 0 for b = 1, 0 for b = 2, and 1 for each b >= 3.

Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, page 60. 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 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 (constant convergence speed), A369771 (stable digits at height 3), A369826 (stable digits at height 2), A370211 (convergence speed at height 3), A371048, A371074 (convergence speed at height 1), A371671 (convergence speed at height 2).

For any n > 1 not a multiple of 10, 1 <= a(n) <= tilde(v(a))+2, where tilde(v(a)) := v_5(a-1) iff a == 1 (mod 5), v_5(a^2+1) iff a == {2, 3} (mod 5), v_5(a+1) iff a == 4 (mod 5), v_2(a^2-1)-1 iff a == 5 (mod 10), where v_2(x) = A007814(x) and v_5(x) = A112765(x) are the 2-adic and 5-adic valuations, respectively (see "Number of stable digits of any integer tetration", p. 447, Definition 2.1, in Links).

A371078 Numbers formed by the rightmost decimal digits of n^n that are the same as those n^(n^n), where -1 indicates that n^n <> n^(n^n) (mod 10).

Original entry on oeis.org

0, 1, -1, 7, 6, 3125, 656, 43, 6, 89, 0, 611, 6, 53, 6, 380859375, 616, 77, -1, 79, 0, 421, -1, 7, 76, 447265625, 203776, 3, 6, 69, 0, 431, 76, 13, 6, 46875, 136, 17, -1, 59, 0, 641, -1, 7, 6, 828125, 696, 3, 6, 449, 0, 15051, 6, 73, 6, 5234375, 456, 688057

Offset: 0

Marco Ripà, Mar 10 2024

The common digits might include leading 0's (such as at n = 5 or n = 43) and they are discarded (in particular, a(0) = 0 indicates that the corresponding zero digit term results in a 0 integer entry).
For n = k*10^c with c >= 1 and k != 0 (mod 10), a(n) = 0 since n^n has (exactly) c*n rightmost 0's.
a(n) is equal to -1 if and only if n == 2 or 18 (mod 20).

			For n = 6, 6^6 = 46656 and 6^(6^6) == 8656 (mod 10^4) so there are three common final digits and a(6) = 656.

Jorge Jiménez Urroz and José Luis Andrés Yebra, On the Equation a^x == x (mod b^n), Journal of Integer Sequences, Article 09.8.8, 2009.
Marco Ripà, Congruence speed of tetration bases ending with 0, arXiv:2402.07929 [math.NT], 2024.

Cf. A000312, A317905, A349425, A369624, A369826, A370211, A371048, A371074.

If n <> 2,18 (mod 20), then a(n) = A000312(n) (mod 10^k), where k is such that n^n == n^(n^n) (mod 10^k) and n^n <> n^(n^n) (mod 10^(k+1)), whereas a(n) = -1 otherwise.

cp's OEIS Frontend

A349425 n-th stable digit (in decimal system) of n^(n^(...^n)).

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A372490 Minimum integer such that the convergence speed of the tetration n^^(a(n)) is constant (i.e., the convergence speed of n^^(a(n) + m) is a fixed number for any positive integer m) and -1 if n is a multiple of 10.

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A371078 Numbers formed by the rightmost decimal digits of n^n that are the same as those n^(n^n), where -1 indicates that n^n <> n^(n^n) (mod 10).

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