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A299241 Ranks of {2,3}-power towers in which #2's = #3's; see Comments.

%I A299241 #15 Aug 07 2024 15:23:12
%S A299241 4,5,18,20,22,23,25,31,62,74,76,77,82,84,85,90,92,93,96,97,99,104,105,
%T A299241 107,128,129,131,135,238,246,250,252,253,294,298,300,301,306,308,309,
%U A299241 312,313,315,326,330,332,333,338,340,341,344,345,347,358,362,364
%N A299241 Ranks of {2,3}-power towers in which #2's = #3's; see Comments.
%C A299241 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.
%C A299241 This sequence together with A299240 and A299242 partition the positive integers.
%H A299241 Clark Kimberling, <a href="/A299241/b299241.txt">Table of n, a(n) for n = 1..1000</a>
%e A299241 The first six terms are the ranks of these towers: t(4) = (2,3), t(5) = (3,2), t(18) = (3,3,2,2), t(20) = (3,2,2,3), t(22) = (3,2,3,2), t(23) = (2,3,2,3).
%t A299241 t[1] = {2}; t[2] = {3}; t[3] = {2, 2}; t[4] = {2, 3}; t[5] = {3, 2};
%t A299241 t[6] = {2, 2, 2}; t[7] = {3, 3}; t[8] = {3, 2, 2}; t[9] = {2, 2, 3};
%t A299241 t[10] = {2, 3, 2}; t[11] = {3, 2, 3}; t[12] = {3, 3, 2};
%t A299241 z = 190; g[k_] := If[EvenQ[k], {2}, {3}]; f = 6;
%t A299241 While[f < 13, n = f; While[n < z, p = 1;
%t A299241   While[p < 12, m = 2 n + 1; v = t[n]; k = 0;
%t A299241     While[k < 2^p, t[m + k] = Join[g[k], t[n + Floor[k/2]]]; k = k + 1];
%t A299241    p = p + 1; n = m]]; f = f + 1]
%t A299241 Select[Range[1000], Count[t[#], 2] > Count[t[#], 3] &];   (* A299240 *)
%t A299241 Select[Range[1000], Count[t[#], 2] == Count[t[#], 3] &]; (* this sequence *)
%t A299241 Select[Range[1000], Count[t[#], 2] < Count[t[#], 3] &];   (* A299242 *)
%Y A299241 Cf. A299229, A299240, A299242.
%K A299241 nonn,easy
%O A299241 1,1
%A A299241 _Clark Kimberling_, Feb 07 2018