A376883 - OEIS

%I #90 Dec 27 2024 10:48:32

%S 8,6,4,5,8,6,4,2,1,4,4,6,8,5,6,4,8,6,6,6,8,4,8,5,6,6,6,6,1,4,2,2,2,5,

%T 4,2,8,8,6,4,8,6,6,5,2,4,2,6,5,5,6,8,6,5,2,2,8,2,1,4,8,4,8,5,6,4,4,4,

%U 6,4,8,4,4,5,8,8,8,4,1,6,8,6,2,5,4,6,8

%N Phase shift of the tetration base n at height n.

%C Let n^^b be n^n^...^n b-times (integer tetration).

%C From here on, we call "stable digits" (or frozen digits) of any given tetration n^^b all and only the rightmost digits of the above-mentioned tetration that matches the corresponding string of right-hand digits generated by the unlimited power tower n^(n^(n^...)).

%C We define as "constant congruence speed" of n all the nonnegative terms of A373387(n).

%C Let #S(n) indicate the total number of the least significant stable digits of n at height n. Additionally, for any n not a multiple of 10, let bar_b be the smallest hyperexponent of the tetration base n such that its congruence speed is constant (see A373387(n)), and assume bar_b = 3 if n is multiple of 10.

%C If n > 1, we note that a noteworthy property of the phase shift of n at any height b >= bar_b is that it describes a cycle whose period length is either 1, 2, or 4 so that (assuming b >= bar_b) the phase shift of n at height b is always equal to the phase shift of n at height b+4, b+8, and so forth.

%C Lastly, for any n, the phase shift of n at height n is defined as the (least significant non-stable digit of n^^n minus the corresponding digit of n^^(n+1)) mod 10 (e.g., the phase shift of 2 at height 2 is (4 - 6) mod 10 = 8).

%C Now, if n > 2 is not a multiple of 10 and is such that A377126(n) = 1, then a(n) = A376842(n) since the congruence speed of n is certainly stable at height n being a sufficient but not necessary condition for the constancy of the congruence speed of n that the hyperexponent of the given base is greater than or equal to 2 + v(n), where v(n) is equal to u_5(n - 1) iff n == 1 (mod 5), u_5(n^2 + 1) iff n == 2,3 (mod 5), u_5(n + 1) iff n == 4 (mod 5), u_2(n^2 - 1) - 1 iff n == 5 (mod 10), while u_5 and u_2 indicate the 5-adic and the 2-adic valuation of the argument (respectively).

%C Since 4 is a multiple of every A377126(n), a(n) is equal to ((n^((n - bar_b) mod 4 + bar_b) - n^((n - bar_b) mod 4 + bar_b + 1))/10^#S(n)) mod 10.

%C Moreover, if n is not a multiple of 10, a(n) is also equal to ((n^((n - (v(n) + 2)) mod 4 + (v(n) + 2)) - n^((n - (v(n) + 2)) mod 4 + (v(n) + 2) + 1))/10^#S(n)) mod 10, where v(n) is equal to

%C u_5(n - 1)        iff n == 1 (mod 5),

%C u_5(n^2 + 1)      iff n == 2,3 (mod 5),

%C u_5(n + 1)        iff n == 4 (mod 5),

%C u_2(n^2 - 1) - 1  iff n == 5 (mod 10) (u_5 and u_2 indicate the 5-adic and the 2-adic valuation of the argument, respectively, see Comments of A373387).

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

%H Marco Ripà, <a href="https://doi.org/10.7546/nntdm.2021.27.4.43-61">The congruence speed formula</a>, Notes on Number Theory and DiscreteMathematics, 2021, 27(4), 43-61.

%H Marco Ripà and Luca Onnis, <a href="https://doi.org/10.7546/nntdm.2022.28.3.441-457">Number of stable digits of any integer tetration</a>, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441-457.

%H Marco Ripà, <a href="https://arxiv.org/abs/2411.00015">Congruence speed of tetration bases ending with 0</a>, arxiv (math.NT), 2024.

%H Marco Ripà, <a href="https://arxiv.org/abs/2411.00015">Graham's number stable digits: an exact solution</a>, arXiv:2411.00015 [math.GM], 2024.

%H Marco Ripà, <a href="https://www.researchgate.net/publication/387314761_Twelve_Python_Programs_to_Help_Readers_Test_Peculiar_Properties_of_Integer_Tetration">Twelve Python Programs to Help Readers Test Peculiar Properties of Integer Tetration</a>, ResearchGate, 2024. See pp. 18, 27.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>.

%F a(n) = ((n^((n - (v(n) + 2)) mod 4 + (v(n) + 2)) - n^((n - (v(n) + 2)) mod 4 + (v(n) + 2) + 1))/10^#S(n)) mod 10 if n not a multiple of 10, and a(n) = A377124(n/10) if n is a multiple of 10.

%e a(11) = 4 since A376842(11) = 4 is a 1 digit number.

%Y Cf. A065881, A317905, A372490, A373387, A376446, A376838, A376842, A377124, A377126.

%K nonn,base,hard

%O 2,1

%A _Marco Ripà_, Oct 25 2024