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%I #15 Apr 03 2024 05:06:17
%S 0,0,0,1,1,1,4,1,2,1,1,9999999990,1,1,1,1,4,1,1,2,1,
%T 104857599999999999999999980,1,1,1,2,3,2,1,1,1,
%U 205891132094648999999999999999999999999999970,1,2,1,1,2,1,1,1,1,12089258196146291747061759999999999999999999999999999999999999960
%N Convergence speed of n at height 3 (i.e., A369771(n) - A369826(n)).
%C 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).
%C 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").
%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.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>.
%F a(n) = A369771(n) - A369826(n).
%e For n = 5, a(n) = 4 since A369771(n) - A369826(n) = 8 - 4.
%Y Cf. A002488, A002489, A317905 (constant convergence speed), A369624, A369771 (n^^3 and n^^4), A369826 (n^^2 and n^^3).
%Y Cf. A007814, A112765.
%K nonn,base
%O -1,7
%A _Marco Ripà_, Feb 11 2024