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A299230 a(n) = height of n-th {2,3}-power tower; see Comments.

%I A299230 #9 Aug 07 2024 11:56:18
%S A299230 1,1,2,2,2,3,2,3,3,3,3,3,4,4,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,4,4,4,4,
%T A299230 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,5,5,5,5,5,5,
%U A299230 5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6
%N A299230 a(n) = height of n-th {2,3}-power tower; see Comments.
%C A299230 Suppose that S is a set of real numbers. An S-power-tower, t, is a number t = x(1)^x(2)^...^x(k), where k >= 1 and x(i) is in S for i = 1..k. We represent t by (x(1), x(2), ..., x(k)), which for k > 1 is defined as (x(1), (x(2), ..., x(k))); (2,3,2) means 2^9. The number k is the *height* of t. If every element of S exceeds 1 and all the power towers are ranked in increasing order, the position of each in the resulting sequence is its *rank*. See A299229 for a guide to related sequences.
%H A299230 Clark Kimberling, <a href="/A299230/b299230.txt">Table of n, a(n) for n = 1..10000</a>
%e A299230 t(8) = (3,2,2), so that a(8) = 3.
%t A299230 t[1] = {2}; t[2] = {3}; t[3] = {2, 2}; t[4] = {2, 3}; t[5] = {3, 2};
%t A299230 t[6] = {2, 2, 2}; t[7] = {3, 3}; t[8] = {3, 2, 2}; t[9] = {2, 2, 3};
%t A299230 t[10] = {2, 3, 2}; t[11] = {3, 2, 3}; t[12] = {3, 3, 2};
%t A299230 z = 190; g[k_] := If[EvenQ[k], {2}, {3}]; f = 6;
%t A299230 While[f < 13, n = f; While[n < z, p = 1;
%t A299230   While[p < 12, m = 2 n + 1; v = t[n]; k = 0;
%t A299230     While[k < 2^p, t[m + k] = Join[g[k], t[n + Floor[k/2]]]; k = k + 1];
%t A299230    p = p + 1; n = m]]; f = f + 1]
%t A299230 Flatten[Table[t[n], {n, 1, 120}]];    (* A299229 *)
%t A299230 w = Table[Length[t[n]], {n, 1, 120}]; (* A299230 *)
%Y A299230 Cf. A299229.
%K A299230 nonn,easy
%O A299230 1,3
%A A299230 _Clark Kimberling_, Feb 06 2018