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

%I

%S 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 47,51,55,56,57,58,59,60,61,63,71,72,73,75,79,80,81,83,87,88,89,91,95,

%U 103,111,112,113,114,115,116,117,118,119,120,121,122,123,124

%N Ranks of {2,3}-power towers in which #2's > #3's; 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-1)); (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 This sequence together with A299241 and A299242 partition the positive integers.

%H Clark Kimberling, <a href="/A299240/b299240.txt">Table of n, a(n) for n = 1..1000</a>

%e 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 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 Select[Range[1000], Count[t[#], 2] > Count[t[#], 3] &]; (* A299240 *)

%t Select[Range[1000], Count[t[#], 2] == Count[t[#], 3] &]; (* A299241 *)

%t Select[Range[1000], Count[t[#], 2] < Count[t[#], 3] &]; (* A299242 *)

%Y Cf. A299229, A299241, A299242.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Feb 07 2018

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