%I #9 Jun 09 2024 23:33:16

%S 25,55,57,68,105,124,126,135,185,193,215,249,265,295,318,345,374,375,

%T 376,425,432,455,505,535,568,585,615,665,682,695,745,751,775,807,818,

%U 825,855,874,876,905,932,935,943,985,999,1001,1015,1057,1065,1095,1124

%N 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).

%C 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).

%H Marco Ripà, <a href="https://doi.org/10.7546/nntdm.2020.26.3.245-260">On the constant congruence speed of tetration</a>, Notes on Number Theory and Discrete Mathematics, Volume 26, 2020, Number 3, Pages 245—260 (see Table 1, pp. 249—251).

%H Marco Ripà and Luca Onnis, <a href="https://doi.org/10.7546/nntdm.2022.28.3.441-457">Number of stable digits of any integer tetration</a>, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441—457 (see Equation 16, p. 454).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>

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

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

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

%K nonn,base

%O 1,1

%A _Marco Ripà_, May 01 2024