A299233 Ranks of {2,3}-power towers that end with 2; see Comments.

1, 3, 5, 6, 8, 10, 12, 13, 14, 17, 18, 21, 22, 25, 26, 27, 28, 29, 30, 35, 36, 37, 38, 43, 44, 45, 46, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 71, 72, 73, 74, 75, 76, 77, 78, 87, 88, 89, 90, 91, 92, 93, 94, 103, 104, 105, 106, 107, 108, 109, 110, 111

Offset

1,2

Comments

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.

Links

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

Example

t(76) = (3,2,3,3,2,2), so that 76 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, A299234 (complement).

Sequence in context: A068125 A139437 A233746 * A320997 A083042 A082977

Adjacent sequences: A299230 A299231 A299232 * A299234 A299235 A299236

Keyword

nonn,easy,changed

Author

Clark Kimberling, Feb 06 2018

Status

approved