A371129 Integers m whose (constant) convergence speed is exactly 3 (i.e., m^^(m+1) has 3 more rightmost frozen digits than m^^m, where ^^ indicates tetration).

25, 55, 57, 68, 105, 124, 126, 135, 185, 193, 215, 249, 265, 295, 318, 345, 374, 375, 376, 425, 432, 455, 505, 535, 568, 585, 615, 665, 682, 695, 745, 751, 775, 807, 818, 825, 855, 874, 876, 905, 932, 935, 943, 985, 999, 1001, 1015, 1057, 1065, 1095, 1124

Offset

1,1

Comments

It is well known (see Links) that as the hyperexponent of the integer m becomes sufficiently large, the constant convergence speed of m is the number of new stable digits that appear at the end of the result for any further unit increment of the hyperexponent itself, and a sufficient (but not necessary) condition to get this fixed value is to set the hyperexponent equal to m plus 1 (e.g., if n := 3, m = 57 and so 57^^58 has exactly 3 more stable digits at the end of the result than 57^^57).

Links

Table of n, a(n) for n=1..51.

Marco Ripà, On the constant congruence speed of tetration, Notes on Number Theory and Discrete Mathematics, Volume 26, 2020, Number 3, Pages 245—260 (see Table 1, pp. 249—251).

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 (see Equation 16, p. 454).

Wikipedia, Tetration

Formula

a(n) is such that A317905(m) = 3, for m = 25, 26, 27, ...

Example

If n = 3, m = 57 and so 57^^58 has exactly 3 more stable digits at the end of the result than 57^^57.

Crossrefs

Cf. A317905 (convergence speed of m^^m), A321130, A321131, A370775.

Sequence in context: A123825 A349416 A157269 * A186892 A206075 A276448

Adjacent sequences: A371126 A371127 A371128 * A371130 A371131 A371132

Keyword

nonn,base

Author

Marco Ripà, May 01 2024

Status

approved