A370211 Convergence speed of n at height 3 (i.e., A369771(n) - A369826(n)).

0, 0, 0, 1, 1, 1, 4, 1, 2, 1, 1, 9999999990, 1, 1, 1, 1, 4, 1, 1, 2, 1, 104857599999999999999999980, 1, 1, 1, 2, 3, 2, 1, 1, 1, 205891132094648999999999999999999999999999970, 1, 2, 1, 1, 2, 1, 1, 1, 1, 12089258196146291747061759999999999999999999999999999999999999960

Offset

-1,7

Comments

A sufficient but not necessary condition for having a constant value of the convergence speed of a tetration base n that is not a multiple of 10 (see A317905) is that the height of the hyperexponent is greater than or equal to tilde(v(a))+2, where tilde(v(a)) := v_5(a-1) iff a == 1 (mod 5), v_5(a^2+1) iff a == {2, 3} (mod 5), v_5(a+1) iff a == 4 (mod 5), v_2(a^2-1)-1 iff a == 5 (mod 10), where v_2(x) = A007814(x) and v_5(x) = A112765(x) are the 2-adic and 5-adic valuations, respectively (see "Number of stable digits of any integer tetration", p. 447, Definition 2.1, in Links).

In detail, considering n > 2 that is not a multiple of 10, a(n) corresponds to the constant convergence speed of the tetration base n, as described by A317905, in almost all the cases since the only term of the provided data of present sequence (i.e., from a(3) to a(40)) that does not match the value of the constant convergence speed of n is a(5) = 4, instead of the correct value of the constant convergence speed of 5 which is v_2(5-1) = 2 (by Equation (16), Line 5, of "Number of stable digits of any integer tetration").

Links

Table of n, a(n) for n=-1..40.

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.

Formula

a(n) = A369771(n) - A369826(n).

Example

For n = 5, a(n) = 4 since A369771(n) - A369826(n) = 8 - 4.

Crossrefs

Cf. A002488, A002489, A317905 (constant convergence speed), A369624, A369771 (n^^3 and n^^4), A369826 (n^^2 and n^^3).

Cf. A007814, A112765.

Sequence in context: A153094 A370050 A144870 * A256252 A247004 A010640

Adjacent sequences: A370208 A370209 A370210 * A370212 A370213 A370214

Keyword

nonn,base

Author

Marco Ripà, Feb 11 2024

Status

approved