A371720 a(n) = m^^m mod 10^len(m), where m = A038399(n) and ^^ indicates tetration or hyper-4.

1, 11, 811, 3811, 63811, 763811, 3763811, 5103763811, 515103763811, 19515103763811, 6819515103763811, 8146819515103763811, 3808146819515103763811, 7213808146819515103763811, 9807213808146819515103763811, 4939807213808146819515103763811

Offset: 1

Author

Marco Ripà, Apr 04 2024

Keywords

• nonn
• base
• more

Comments

For any n, a(n) == a(n + 1) (mod 10^len(A038399(n))), where len(k) := number of digits in k. Assuming len(a(n)) > 1, this is a general property of every concatenated sequence with fixed rightmost digits (such as A014925, A061839, A092845, and A104759), as shown in Ripà's book "La strana coda della serie n^n^...^n".

Moreover, assuming n > 1, since A038399(n) is congruent to 11 (mod 20), the convergence speed of A038399(n)^^b (say, V(A038399(n), b) = {2, 1, 1, 1, ...}) is 2 at height 1 and becomes a unit value for any integer b > 1 (see Links). Hence, a(n) is given by A038399(n)^^len(A038399(n) - 1) (mod 10^len(A038399(n))), and also by A038399(n)^^len(A038399(n)) (mod 10^len(A038399(n))) since A038399(n)^^len(A038399(n)) == A038399(n)^^len(A038399(n) - 1) (mod 10^len(A038399(n))) holds for any n.

Examples

			a(8) is given by the rightmost 10 digits of 2113853211^^2113853211 and thus a(8) = 5103763811.
a(9) == a(8) (mod 10^10), i.e., the digits of a(9) end with the digits of a(8) (and then a(9) has 2 more preceding).
		

References

• Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, page 60. 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 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.

Crossrefs

Cf. A000045 (Fibonacci), A038399, A171882 (tetration), A317824, A317903, A317905.

Formula

a(n) = A038399(n)^^(len(A038399(n)) - 1) mod 10^len(A038399(n)), where len(A038399(n)) = ceiling(log_10(A038399(n) + 1)).