login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

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

%I #9 Aug 07 2024 11:56:57

%S 1,3,5,6,8,10,12,13,14,17,18,21,22,25,26,27,28,29,30,35,36,37,38,43,

%T 44,45,46,51,52,53,54,55,56,57,58,59,60,61,62,71,72,73,74,75,76,77,78,

%U 87,88,89,90,91,92,93,94,103,104,105,106,107,108,109,110,111

%N Ranks of {2,3}-power towers that end 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="/A299233/b299233.txt">Table of n, a(n) for n = 1..1000</a>

%e t(76) = (3,2,3,3,2,2), so that 76 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], Last[t[#]] == 2 &]; (* A299233 *)

%t Select[Range[200], Last[t[#]] == 3 &]; (* A299234 *)

%Y Cf. A299229, A299234 (complement).

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Feb 06 2018