A317903 -id:A317903 - cp's OEIS frontend

A318478 Decimal digits such that for all k>=1, the number A(k) := Sum_{n = 0..k-1 } a(n)*10^n satisfies the congruence 1984^A(k) == A(k) (mod 10^k).

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

Offset: 1

Marco Ripà, Aug 26 2018

10-adic expansion of the iterated exponential 1984^^n for sufficiently large n (where c^^n denotes a tower of c's of height n). E.g., for n>=9, 1984^^n(mod 10^8) == 98703616.

1984^^n, for any n>=188, appears in M. Ripà's book "La strana coda della serie n^n^...^n", where the author took his birth year (1984), as a random base in order to prove some general properties about tetration, and calculating 1984^^n(mod 10^187) as a test for his paper-and-pencil procedure.

			1984^^1984 (mod 10^8) == 98703616.
Thus, 1984^^1984 = ...61630789307145912032948400109045102(...)7490335.
Consider the sequence 1984^^n: 1984, 1984^1984, 1984^(1984^1984), ... From 1984^^3 onwards, all terms end with the digits 16. This follows from Euler's generalization of Fermat's little theorem.

M. Gardner, Mathematical Games, Scientific American 237, 18 - 28 (1977).
M. Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, p. 78-79. ISBN 978-88-6178-789-6.
Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, pages 226-229.

J. Jimenez Urroz and J. Luis A. Yebra, On the equation a^x == x (mod b^n), J. Int. Seq. 12 (2009) #09.8.8.
Robert P. Munafo, Large Numbers
Wikipedia, Graham's number
Wikipedia, Tetration

Cf. A133612, A133613, A133614, A133615, A133616, A133617, A133618, A133619, A144539, A144540, A144541, A144542, A144543, A144544, A317824, A317903, A317905.

A371720 a(n) = m^^m mod 10^len(m), where m = A038399(n) and ^^ indicates tetration or hyper-4.

Original entry on oeis.org

1, 11, 811, 3811, 63811, 763811, 3763811, 5103763811, 515103763811, 19515103763811, 6819515103763811, 8146819515103763811, 3808146819515103763811, 7213808146819515103763811, 9807213808146819515103763811, 4939807213808146819515103763811

Offset: 1

Marco Ripà, Apr 04 2024

For any n, a(n) == a(n + 1) (mod 10^len(A038399(n))), where len(k) := number of digits in k. Assuming len(a(n)) > 1, this is a general property of every concatenated sequence with fixed rightmost digits (such as A014925, A061839, A092845, and A104759), as shown in Ripà's book "La strana coda della serie n^n^...^n".

Moreover, assuming n > 1, since A038399(n) is congruent to 11 (mod 20), the convergence speed of A038399(n)^^b (say, V(A038399(n), b) = {2, 1, 1, 1, ...}) is 2 at height 1 and becomes a unit value for any integer b > 1 (see Links). Hence, a(n) is given by A038399(n)^^len(A038399(n) - 1) (mod 10^len(A038399(n))), and also by A038399(n)^^len(A038399(n)) (mod 10^len(A038399(n))) since A038399(n)^^len(A038399(n)) == A038399(n)^^len(A038399(n) - 1) (mod 10^len(A038399(n))) holds for any n.

			a(8) is given by the rightmost 10 digits of 2113853211^^2113853211 and thus a(8) = 5103763811.
a(9) == a(8) (mod 10^10), i.e., the digits of a(9) end with the digits of a(8) (and then a(9) has 2 more preceding).

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. A000045 (Fibonacci), A038399, A171882 (tetration), A317824, A317903, A317905.

a(n) = A038399(n)^^(len(A038399(n)) - 1) mod 10^len(A038399(n)), where len(A038399(n)) = ceiling(log_10(A038399(n) + 1)).

A386005 Number of stable digits of the integer tetration n^^n (i.e., maximum nonnegative integer m such that n^^n is congruent modulo 10^m to n^^(n + 1)), or -1 if n is a multiple of 10.

Original entry on oeis.org

0, 2, 3, 12, 7, 12, 7, 9, -1, 12, 11, 13, 13, 61, 17, 17, 32, 19, -1, 22, 20, 22, 46, 78, 54, 26, 27, 29, -1, 32, 62, 33, 33, 71, 37, 37, 36, 39, -1, 42, 40, 84, 43, 92, 47, 46, 47, 98, -1, 103, 51, 53, 53, 166, 57, 171, 56, 59, -1, 62, 60, 62, 63, 396, 67, 66

Offset: 2

Marco Ripà, Jul 14 2025

If n is divisible by 10, the corresponding a(n) would be too large to be include in the data section (it would be equal to the number of trailing zeros that appear at the end of n^^n).

This sequence directly follows from the constancy of the "congruence speed" property characterizing tetration (for an explicit formula to calculate a(n) for any n not a multiple of 10, see the linked paper entitled "Number of stable digits of any integer tetration").

			For n = 5, 5^5^5^5^5 is congruent to 5^5^5^5^5^5 (mod 10^12) and 5^5^5^5^5 is not congruent to 5^5^5^5^5^5 (mod 10^13). Thus, a(5) = 12.

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 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. A317824, A317903, A317905, A349425, A356946.

def last_k_digits_tetration(base, height, k):
    mod = 10 ** k
    result = base
    for _ in range(height - 1):
        result = pow(base, result, mod)
    return str(result).zfill(k)
def count_stable_digits(base, k=500):
    try:
        x = last_k_digits_tetration(base, base, k)
        y = last_k_digits_tetration(base, base + 1, k)
        count = 0
        for i in range(1, k + 1):
            if x[-i] == y[-i]:
                count += 1
            else:
                break
        return count
    except:
        return -1
for n in range(2, 101):
    if n % 10 == 0:
        print(f"n = {n}: -1")
    else:
        print(f"n = {n}: {count_stable_digits(n)}")

If n == 0 (mod 10), then a(n) = -1, and a(n) = A356946(n) otherwise.

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A318478 Decimal digits such that for all k>=1, the number A(k) := Sum_{n = 0..k-1 } a(n)*10^n satisfies the congruence 1984^A(k) == A(k) (mod 10^k).

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A371720 a(n) = m^^m mod 10^len(m), where m = A038399(n) and ^^ indicates tetration or hyper-4.

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A386005 Number of stable digits of the integer tetration n^^n (i.e., maximum nonnegative integer m such that n^^n is congruent modulo 10^m to n^^(n + 1)), or -1 if n is a multiple of 10.

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