A299232 Ranks of {2,3}-power towers that start with 3; see Comments.

2, 5, 7, 8, 11, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124

Offset

1,1

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..10000

Formula

a(n) = 2n for all n except 2, 3, and 5.

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], First[t[#]] == 2 &]; (* A299231 *)
Select[Range[200], First[t[#]] == 3 &]; (* A299232 *)

Crossrefs

Cf. A299229, A299231 (complement).

Sequence in context: A331866 A242401 A050941 * A153083 A140592 A153086

Adjacent sequences: A299229 A299230 A299231 * A299233 A299234 A299235

Keyword

nonn,easy

Author

Clark Kimberling, Feb 06 2018

Status

approved