A356946 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)).

1, 0, 2, 3, 12, 7, 12, 7, 9

Offset

1,3

Comments

a(10) = 10^^9 is too large to include. In general, if n is a multiple of 10, then a(n) is given by the number of trailing zeros that appear at the end of n^^n.

This follows from the constancy of the "congruence speed" (AKA "convergence speed" here on the OEIS) of hyper-3 for any exponentiation base which is a multiple of 10, otherwise the congruence speed is constant only for hyper-4 and it is strictly positive for any tetration base n >= 1 that is not a multiple of 10 (for an explicit formula to calculate a(n) for any n, see the linked paper entitled "Number of stable digits of any integer tetration").

References

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

Links

Table of n, a(n) for n=1..9.

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

Example

For n = 3, 3^3^3 is congruent to 3^3^3^3 (mod 10^2) and 3^3^3 is not congruent to 3^3^3^3 (mod 10^3). Thus, a(3) = 2.

Crossrefs

Cf. A317824, A317903, A317905, A349425.

Sequence in context: A292265 A259416 A137765 * A287509 A282417 A287607

Adjacent sequences: A356943 A356944 A356945 * A356947 A356948 A356949

Keyword

nonn,base

Author

Marco Ripà, Sep 05 2022

Status

approved