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A299231 Ranks of {2,3}-power towers that start with 2; see Comments.

%I A299231 #9 Aug 07 2024 11:57:08
%S A299231 1,3,4,6,9,10,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,
%T A299231 49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,
%U A299231 95,97,99,101,103,105,107,109,111,113,115,117,119,121,123,125
%N A299231 Ranks of {2,3}-power towers that start with 2; see Comments.
%C A299231 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 A299231 Clark Kimberling, <a href="/A299231/b299231.txt">Table of n, a(n) for n = 1..10000</a>
%F A299231 a(n) = 2n-1 for all n except 3, 4, and 6.
%e A299231 t(97) = (2,3,2,3,2,3), so that 97 is in the sequence.
%t A299231 t[1] = {2}; t[2] = {3}; t[3] = {2, 2}; t[4] = {2, 3}; t[5] = {3, 2};
%t A299231 t[6] = {2, 2, 2}; t[7] = {3, 3}; t[8] = {3, 2, 2}; t[9] = {2, 2, 3};
%t A299231 t[10] = {2, 3, 2}; t[11] = {3, 2, 3}; t[12] = {3, 3, 2};
%t A299231 z = 190; g[k_] := If[EvenQ[k], {2}, {3}]; f = 6;
%t A299231 While[f < 13, n = f; While[n < z, p = 1;
%t A299231   While[p < 12, m = 2 n + 1; v = t[n]; k = 0;
%t A299231     While[k < 2^p, t[m + k] = Join[g[k], t[n + Floor[k/2]]]; k = k + 1];
%t A299231    p = p + 1; n = m]]; f = f + 1]
%t A299231 Select[Range[200], First[t[#]] == 2 &]; (* A299231 *)
%t A299231 Select[Range[200], First[t[#]] == 3 &]; (* A299232 *)
%Y A299231 Cf. A299229, A299232 (complement).
%K A299231 nonn,easy
%O A299231 1,2
%A A299231 _Clark Kimberling_, Feb 06 2018