A377126 Number of digits of A376842(n) or -1 if A376842(n) = -1.

1, 2, 4, 1, 4, 4, 4, 1, -1, 1, 2, 4, 1, 1, 2, 1, 2, 4, -1, 4, 4, 4, 1, 1, 2, 4, 4, 4, -1, 4, 4, 2, 2, 1, 1, 4, 2, 2, -1, 2, 4, 2, 4, 1, 4, 1, 1, 4, -1, 1, 1, 1, 4, 1, 4, 4, 4, 1, -1, 4, 2, 4, 1, 1, 2, 2, 4, 4, -1, 4, 4, 4, 4, 1, 4, 4, 4, 4, -1, 4, 4, 4, 4, 1

Offset: 2

Author

Marco Ripà, Oct 17 2024

Keywords

• sign
• base

Comments

For any integer n > 1 not a multiple of 10, a(n) belongs to the set {1, 2, 4}. Furthermore, if the last digit of n is 5, then A376446(n) = 5 so that a(n) = 1. Conversely, by definition, a(n) -1 if and only if n is congruent to 0 modulo 10.

This sequence is also equal to the number of digits of A376446(n) and -1 if A376446(n) = -1; for the values of the phase shifts at heights 2 and 3 of any tetration base n which is a multiple of 10, see A376838 and A377124 (respectively).

Examples

			a(4) = 4 since A376446(4) = 2486 is a 4 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, On the constant congruence speed of tetration, Notes on Number Theory and Discrete Mathematics, Volume 26, 2020, Number 3, Pages 245-260.
• 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.
• 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. 21, 27.
• Wikipedia, Tetration.

Crossrefs

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

Formula

a(n) = floor(log_(10)(A376842(n))) + 1.

a(n) = floor(log_(10)(A376446(n))) + 1.

a(n) = -1 iff A376446(n) = -1; a(n) = 1 iff 1 <= A376446(n) <= 9; a(n) = 2 iff A376446(n) = {19, 28, 37, 46, 64, 73, 82, 91}; a(n) = 4 otherwise.