A299229 {2,3}-power towers in increasing order, concatenated; see Comments.

2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 3, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 3, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 2, 3, 2, 2, 3, 3, 2, 2, 2, 2, 2, 3, 3, 2, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 3, 2, 3, 3, 3, 2, 2, 2, 2, 2

Offset: 1

Author

Clark Kimberling, Feb 06 2018

Keywords

• nonn
• easy
• tabf

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*.

In the following guide, "tower" means "power-tower", and t(n) denotes the n-th {2,3}-tower, represented as (x(1), x(2), ..., x(k)).

A299229: sequence of all {2,3}-towers, ranked, concatenated

A299230: a(n) = height of t(n)

A299231: all n such that t(n) has x(1) = 2

A299232: all n such that t(n) has x(1) = 3

A299233: all n such that t(n) has x(k) = 2

A299234: all n such that t(n) has x(k) = 3

A299235: a(n) = number of 2's in t(n)

A299236: a(n) = number of 3's in t(n)

A299237: a(n) = m satisfying t(m) = reversal of t(n)

A299238; a(n) = m satisfying t(m) = 5 - t(n)

A299239: all n such that t(n) is a palindrome

A299240: ranks of all t[n] in which #2's > #3's

A299241: ranks of all t[n] in which #2's = #3's

A299242: ranks of all t[n] in which #2's < #3's

A299322: ranks of t[n] in which the 2's and 3's alternate

Rectangular arrays:

A299323: row n shows ranks of towers in which #2's = n

A299324: row n shows ranks of towers in which #3's = n

A299325: row n shows ranks of towers that start with n 2's

A299326: row n shows ranks of towers that start with n 3's

A299327: row n shows ranks of towers having maximal runlength n

Examples

			As an irregular triangle, where row n contains the digits of A248907(n):
  2;
  3;
  2, 2;
  2, 3;
  3, 2;
  2, 2, 2;
  3, 3;
  3, 2, 2;
  2, 2, 3;
  2, 3, 2;
  3, 2, 3;
  3, 3, 2;
  2, 2, 2, 2;
  3, 2, 2, 2;
  2, 3, 3;
  ...
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A248907, A299230-A299242, A256231, A185969, A299322-A299327.

Programs

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]
Flatten[Table[t[n], {n, 1, 120}]];  (* A299229 *)