A299239 - OEIS

%I #11 Aug 07 2024 13:12:57

%S 1,2,3,6,7,10,11,13,16,20,25,27,34,35,40,45,48,53,55,66,70,75,80,89,

%T 100,109,111,119,130,142,147,155,160,168,177,185,196,204,213,221,223,

%U 247,258,266,278,286,291,315,320,344,353,377,388,412,421,445,447

%N Ranks of palindromic {2,3}-power towers; 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="/A299239/b299239.txt">Table of n, a(n) for n = 1..1000</a>

%e The first six palindromes are t(1) = (2), t(2) = (3), t(3) = (2,2), t(6) = (2,2,2), t(7) = (3,3), 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 Flatten[Table[Select[Range[1000], t[#] == Reverse[t[#]] &], {n, 1, 120}]]

%Y Cf. A299229, A299237.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Feb 07 2018