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A377124
Phase shift (original name "sfasamento") of the tetration base 10*n at any height greater than or equal to 3.
1
1, 6, 1, 6, 5, 6, 1, 6, 1, 1, 1, 6, 1, 6, 5, 6, 1, 6, 1, 6, 1, 6, 1, 6, 5, 6, 1, 6, 1, 1, 1, 6, 1, 6, 5, 6, 1, 6, 1, 6, 1, 6, 1, 6, 5, 6, 1, 6, 1, 5, 1, 6, 1, 6, 5, 6, 1, 6, 1, 6, 1, 6, 1, 6, 5, 6, 1, 6, 1, 1, 1, 6, 1, 6, 5, 6, 1, 6, 1, 6, 1, 6, 1, 6, 5, 6, 1
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.
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.
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