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A299232 Ranks of {2,3}-power towers that start with 3; see Comments.

%I A299232 #9 Aug 07 2024 12:10:48
%S A299232 2,5,7,8,11,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,
%T A299232 50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,
%U A299232 96,98,100,102,104,106,108,110,112,114,116,118,120,122,124
%N A299232 Ranks of {2,3}-power towers that start with 3; see Comments.
%C A299232 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 A299232 Clark Kimberling, <a href="/A299232/b299232.txt">Table of n, a(n) for n = 1..10000</a>
%F A299232 a(n) = 2n for all n except 2, 3, and 5.
%e A299232 t(76) = (3,2,3,3,2,2), so that 76 is in the sequence.
%t A299232 t[1] = {2}; t[2] = {3}; t[3] = {2, 2}; t[4] = {2, 3}; t[5] = {3, 2};
%t A299232 t[6] = {2, 2, 2}; t[7] = {3, 3}; t[8] = {3, 2, 2}; t[9] = {2, 2, 3};
%t A299232 t[10] = {2, 3, 2}; t[11] = {3, 2, 3}; t[12] = {3, 3, 2};
%t A299232 z = 190; g[k_] := If[EvenQ[k], {2}, {3}]; f = 6;
%t A299232 While[f < 13, n = f; While[n < z, p = 1;
%t A299232   While[p < 12, m = 2 n + 1; v = t[n]; k = 0;
%t A299232     While[k < 2^p, t[m + k] = Join[g[k], t[n + Floor[k/2]]]; k = k + 1];
%t A299232    p = p + 1; n = m]]; f = f + 1]
%t A299232 Select[Range[200], First[t[#]] == 2 &]; (* A299231 *)
%t A299232 Select[Range[200], First[t[#]] == 3 &]; (* A299232 *)
%Y A299232 Cf. A299229, A299231 (complement).
%K A299232 nonn,easy
%O A299232 1,1
%A A299232 _Clark Kimberling_, Feb 06 2018