A299231 - OEIS

%I #9 Aug 07 2024 11:57:08

%S 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 49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,

%U 95,97,99,101,103,105,107,109,111,113,115,117,119,121,123,125

%N Ranks of {2,3}-power towers that start with 2; 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))); (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 Clark Kimberling, <a href="/A299231/b299231.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = 2n-1 for all n except 3, 4, and 6.

%e t(97) = (2,3,2,3,2,3), so that 97 is in the sequence.

%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[200], First[t[#]] == 2 &]; (* A299231 *)

%t Select[Range[200], First[t[#]] == 3 &]; (* A299232 *)

%Y Cf. A299229, A299232 (complement).

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Feb 06 2018