A376883 Phase shift of the tetration base n at height n.

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, 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, 6, 4, 8, 4, 4, 5, 8, 8, 8, 4, 1, 6, 8, 6, 2, 5, 4, 6, 8

Offset: 2

Author

Marco Ripà, Oct 25 2024

Keywords

• nonn
• base
• hard

Comments

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

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^...)).

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

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.

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.

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).

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).

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.

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

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) (u_5 and u_2 indicate the 5-adic and the 2-adic valuation of the argument, respectively, see Comments of A373387).

Examples

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

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 DiscreteMathematics, 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.
• Marco Ripà, Congruence speed of tetration bases ending with 0, arxiv (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, 27.
• Wikipedia, Tetration.

Crossrefs

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

Formula

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.