A299234 Ranks of {2,3}-power towers that end with 3; see Comments.

2, 4, 7, 9, 11, 15, 16, 19, 20, 23, 24, 31, 32, 33, 34, 39, 40, 41, 42, 47, 48, 49, 50, 63, 64, 65, 66, 67, 68, 69, 70, 79, 80, 81, 82, 83, 84, 85, 86, 95, 96, 97, 98, 99, 100, 101, 102, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140

Offset

1,1

Comments

Suppose that S is a set of real numbers. As 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.

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

Example

t(80) = (3,2,2,2,2,3), so that 80 is in the sequence.

Mathematica

t[1] = {2}; t[2] = {3}; t[3] = {2, 2}; t[4] = {2, 3}; t[5] = {3, 2};
t[6] = {2, 2, 2}; t[7] = {3, 3}; t[8] = {3, 2, 2}; t[9] = {2, 2, 3};
t[10] = {2, 3, 2}; t[11] = {3, 2, 3}; t[12] = {3, 3, 2};
z = 190; g[k_] := If[EvenQ[k], {2}, {3}]; f = 6;
While[f < 13, n = f; While[n < z, p = 1;
  While[p < 12, m = 2 n + 1; v = t[n]; k = 0;
    While[k < 2^p, t[m + k] = Join[g[k], t[n + Floor[k/2]]]; k = k + 1];
   p = p + 1; n = m]]; f = f + 1]
Select[Range[200], Last[t[#]] == 2 &]; (* A299233 *)
Select[Range[200], Last[t[#]] == 3 &]; (* A299234 *)

Crossrefs

Cf. A299229, A299233 (complement).

Sequence in context: A182761 A329830 A081841 * A366462 A213273 A027904

Adjacent sequences: A299231 A299232 A299233 * A299235 A299236 A299237

Keyword

nonn,easy

Author

Clark Kimberling, Feb 06 2018

Status

approved