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a(n) = height of n-th {2,3}-power tower; see Comments.
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%I #9 Aug 07 2024 11:56:18

%S 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 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 5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6

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

%C 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 Clark Kimberling, <a href="/A299230/b299230.txt">Table of n, a(n) for n = 1..10000</a>

%e t(8) = (3,2,2), so that a(8) = 3.

%t t[1] = {2}; t[2] = {3}; t[3] = {2, 2}; t[4] = {2, 3}; t[5] = {3, 2};

%t t[6] = {2, 2, 2}; t[7] = {3, 3}; t[8] = {3, 2, 2}; t[9] = {2, 2, 3};

%t t[10] = {2, 3, 2}; t[11] = {3, 2, 3}; t[12] = {3, 3, 2};

%t z = 190; g[k_] := If[EvenQ[k], {2}, {3}]; f = 6;

%t While[f < 13, n = f; While[n < z, p = 1;

%t While[p < 12, m = 2 n + 1; v = t[n]; k = 0;

%t While[k < 2^p, t[m + k] = Join[g[k], t[n + Floor[k/2]]]; k = k + 1];

%t p = p + 1; n = m]]; f = f + 1]

%t Flatten[Table[t[n], {n, 1, 120}]]; (* A299229 *)

%t w = Table[Length[t[n]], {n, 1, 120}]; (* A299230 *)

%Y Cf. A299229.

%K nonn,easy

%O 1,3

%A _Clark Kimberling_, Feb 06 2018