A299237 a(n) = the index m satisfying t(m) = reversal of t(n), where t(n) is the n-th {2,3}-power tower; see Comments.

1, 2, 3, 5, 4, 6, 7, 9, 8, 10, 11, 15, 13, 19, 12, 16, 21, 31, 14, 20, 17, 23, 22, 32, 25, 33, 27, 39, 43, 63, 18, 24, 26, 34, 35, 47, 51, 67, 28, 40, 44, 64, 29, 41, 45, 65, 36, 48, 52, 68, 37, 49, 53, 69, 55, 79, 87, 127, 71, 95, 103, 135, 30, 42, 46, 66

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.

This sequence is a self-inverse permutation of the positive integers.

Links

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

Example

t(12) = (3,3,2) and t(15) = (2,3,3) = reversal of t(12); therefore a(12) = 15.

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]
r[n_] := Reverse[t[n]]
Flatten[Table[Select[Range[2000], t[#] == r[n] &], {n, 1, 1500}]]; (* A299237 *)

Crossrefs

Cf. A299229, A299239 (fixed points of the permutation; palindromes).

Sequence in context: A131141 A130981 A074145 * A257676 A257677 A105363

Adjacent sequences: A299234 A299235 A299236 * A299238 A299239 A299240

Keyword

nonn,easy

Author

Clark Kimberling, Feb 07 2018

Status

approved