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A299240 Ranks of {2,3}-power towers in which #2's > #3's; see Comments.

%I A299240 #10 Aug 07 2024 11:55:29
%S A299240 1,3,6,8,9,10,13,14,17,19,21,27,28,29,30,35,36,37,39,40,41,43,44,45,
%T A299240 47,51,55,56,57,58,59,60,61,63,71,72,73,75,79,80,81,83,87,88,89,91,95,
%U A299240 103,111,112,113,114,115,116,117,118,119,120,121,122,123,124
%N A299240 Ranks of {2,3}-power towers in which #2's > #3's; see Comments.
%C A299240 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 A299240 This sequence together with A299241 and A299242 partition the positive integers.
%H A299240 Clark Kimberling, <a href="/A299240/b299240.txt">Table of n, a(n) for n = 1..1000</a>
%e A299240 The first six terms are the ranks of these towers: t(1) = (2), t(3) = (2,2), t(6) = (2,2,2), t(8) = (3,2,2), t(9) = (2,2,3), t(10) = (2,3,2).
%t A299240 t[1] = {2}; t[2] = {3}; t[3] = {2, 2}; t[4] = {2, 3}; t[5] = {3, 2};
%t A299240 t[6] = {2, 2, 2}; t[7] = {3, 3}; t[8] = {3, 2, 2}; t[9] = {2, 2, 3};
%t A299240 t[10] = {2, 3, 2}; t[11] = {3, 2, 3}; t[12] = {3, 3, 2};
%t A299240 z = 190; g[k_] := If[EvenQ[k], {2}, {3}]; f = 6;
%t A299240 While[f < 13, n = f; While[n < z, p = 1;
%t A299240   While[p < 12, m = 2 n + 1; v = t[n]; k = 0;
%t A299240     While[k < 2^p, t[m + k] = Join[g[k], t[n + Floor[k/2]]]; k = k + 1];
%t A299240    p = p + 1; n = m]]; f = f + 1]
%t A299240 Select[Range[1000], Count[t[#], 2] > Count[t[#], 3] &];   (* A299240 *)
%t A299240 Select[Range[1000], Count[t[#], 2] == Count[t[#], 3] &];  (* A299241 *)
%t A299240 Select[Range[1000], Count[t[#], 2] < Count[t[#], 3] &];   (* A299242 *)
%Y A299240 Cf. A299229, A299241, A299242.
%K A299240 nonn,easy
%O A299240 1,2
%A A299240 _Clark Kimberling_, Feb 07 2018