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A379243 a(n) = (10^(n + 1) + 10^(n - min{v_2(n), v_5(n)}) + 1)^n, where v_p(n) indicates the p-adic valuation of n.

%I A379243 #10 Dec 30 2024 17:21:20
%S A379243 111,1212201,1331363033001,146415324072600440001,
%T A379243 1610517320513310012100005500001,
%U A379243 1771561966306219615026620001815000066000001,194871722400927338207105124350046585000254100000770000001,2143588825589736849603708090188560102487000074536000033880000008800000001
%N A379243 a(n) = (10^(n + 1) + 10^(n - min{v_2(n), v_5(n)}) + 1)^n, where v_p(n) indicates the p-adic valuation of n.
%C A379243 For any n, a(n) == 1 (mod 10^n), while it is not congruent to 1 modulo 10^(n + 1).
%C A379243 If n is not a multiple of 10, 10^(n + 1) + 10^(n - min{v_2(n), v_5(n)}) + 1 has a total of n + 2 digits and they are n - 1 0s and 3 1s. Conversely, if there is a pair of positive integers (m, k) not ending with 0 and such that n := m*10^k, then 10^(n + 1) + 10^(n - min{v_2(n), v_5(n)}) + 1 has n - k + 2 digits (three 1s and the rest are all 0s) and  a(n) = (10^(n + 1) + 10^(n - k) + 1)^n.
%C A379243 Since a(n) == 1 (mod 5) for any n, the constant congruence speed of a(n) (i.e., V(a(n))) is guaranteed to be constant starting from height v_5(a(n) - 1) + 2 (for this sufficient condition, see “Number of stable digits of any integer tetration” in Links).
%C A379243 Then, for any positive integer n, a(n) is (exactly) a n-th perfect power (since, for any given n, 10^(n + 1) + 10^(n - min{v_2(n), v_5(n)}) + 1 is divisible by 3 only once) and is also characterized by a constant congruence speed of n (for a strict proof of the general formula V((10^(t + k) + 10^(t - min{v_2(n), v_5(n)}) + 1)^n) = t, holding for any chosen positive integer k as long as t is an integer above min{v_2(n), v_5(n)} + 1, see Section 3 of “On the relation between perfect powers and tetration frozen digits” in Links).
%H A379243 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 A379243 Marco Ripà, <a href="https://doi.org/10.59400/jam1771">On the relation between perfect powers and tetration frozen digits</a>, Journal of AppliedMath, 2024, 2(5), 1771.
%H A379243 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>.
%F A379243 If n <> 0 (mod 10), then a(n) = (11...[n - 1 trailing 0s]...1)^n.
%e A379243 a(3) = (10^4 + 10^3 + 1)^3 = 11001^3  = 1331363033001 is a perfect cube whose constant congruence speed is 3.
%t A379243 pAdicValuation[n_, p_] := Module[{v = 0, k = n}, While[Mod[k, p] == 0 && k > 0,k = k/p;v++;];v];
%t A379243 a[n_] := Module[{v2, v5, minVal}, v2 = pAdicValuation[n, 2]; v5 = pAdicValuation[n, 5];
%t A379243 minVal = Min[v2, v5];(10^(n + 1) + 10^(n - minVal) + 1)^n]; sequence = Table[a[n], {n, 1, 20}]; sequence
%Y A379243 Cf. A000533, A001597, A063006, A121520, A199691, A317905, A372490, A373387.
%K A379243 nonn,base
%O A379243 1,1
%A A379243 _Marco Ripà_, Dec 18 2024