OFFSET
1,2
COMMENTS
Let m^^b be m^m^...^m b-times (integer tetration).
For any n, the phase shift of n*10 at height b is defined as the congruence class modulo 10 of the difference between the least significant non-stable digit of (n*10)^^b and the corresponding digit of (n*10)^^(b+1), so the phase shift of n*10 at height 1 is trivially A065881(n) while the phase shift of n*10 at height 2 is given by A376838(n).
Thus, assume b >= 3 and, for any given tetration base n*10, this sequence represents the congruence classes modulo 10 of the differences between the rightmost non-stable digit of (n*10)^^b and the zero of (n*10)^^(b+1) which occupies the same decimal position (counting from right to the left) as the rightmost nonzero digit of (n*10)^^b (see Appendix of "Graham's number stable digits: an exact solution" in Links).
If n == 3,7 (mod 10), a(n) <> A065881(n) since the least significant nonzero digit of (n*10)^^b only depends on the last digit of n^^(b - 1) and, in the mentioned two cases, n*10 is not congruent to 0 modulo 4, whereas (n*10)^(n*10) is clearly a multiple of 4 given the fact that it is also a multiple of 100 (e.g., if n = 3 is given, the last nonzero digit of (n*10)^(n*10) is 3 iff (n*10) == 1 (mod 4), 9 iff (n*10) == 2 (mod 4), 7 iff (n*10) == 3 (mod 4), 1 iff (n*10) == 0 (mod 4), which is the only case we are considering here since (3*10)^(3*10) == 0 (mod 100)).
REFERENCES
Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011. ISBN 978-88-6178-789-6.
LINKS
Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43—61.
Marco Ripà, Congruence speed of tetration bases ending with 0, arXiv:2402.07929 [math.NT], 2024.
Marco Ripà, Graham's number stable digits: an exact solution, arXiv:2411.00015 [math.GM], 2024.
Marco Ripà, Twelve Python Programs to Help Readers Test Peculiar Properties of Integer Tetration, ResearchGate, 2024. See pp. 18, 19, 20, 27.
Wikipedia, Graham's Number.
Wikipedia, Tetration.
FORMULA
a(n) equals the least significant nonzero digit of n^((n*10)^(n*10)).
Let h indicate the least significant nonzero digit of n. Then,
a(n) = 1 iff h = 1,3,7,9;
a(n) = 5 iff h = 5;
a(n) = 6 iff h = 2,4,6,8.
EXAMPLE
a(1) = 1 since 10^(10^10) == 0 (mod 10^10000000000) and 10^(10^10) == 1 (mod 10^10000000001), and trivially 1 - 0 = 1.
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
Marco Ripà, Oct 17 2024
STATUS
approved