A256231 -id:A256231 - cp's OEIS frontend

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

Clark Kimberling, Feb 06 2018

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

			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;
  ...

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

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

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 *)

A185969 Let S be the sequence of power towers built of 2 and 3 sorted by their height and for equal heights - in lexicographic order: 2, 3, 2^2, 2^3, 3^2, 3^3, 2^2^2, 2^2^3 etc. A(n) = the permutation of indexes which reorders S by magnitude.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 11, 8, 9, 12, 13, 15, 23, 10, 14, 19, 27, 16, 24, 17, 25, 20, 28, 21, 29, 31, 47, 39, 55, 18, 26, 22, 30, 35, 51, 43, 59, 32, 48, 40, 56, 33, 49, 41, 57, 36, 52, 44, 60, 37, 53, 45, 61, 63, 95, 79, 111, 71, 103, 87, 119, 34, 50, 42, 58, 38

Offset: 1

Vladimir Reshetnikov, Feb 07 2011

			a(6) =  7; tower(7)  = 2^2^2 = 2^4 =  16.
a(7) =  6; tower(6)  = 3^3   =        27.
a(8) = 11; tower(11) = 3^2^2 = 3^4 =  81.
a(9) =  8; tower(8)  = 2^2^3 = 2^8 = 256.

Alois P. Heinz, Table of n, a(n) for n = 1..14335
Vladimir Reshetnikov, 2-3 sequence puzzle, SeqFan list, Mar 18 2015
Index entries for sequences that are permutations of the natural numbers

Cf. A032810, A081241, A248907, A256179, A256231, A375374 (colexicographic instead of lexicographic order).

a(2*n-1) = A081241(2*A081241(a(n-1))+1) and a(2*n) = A081241(A081241(a(2*n-1))+1) for n >= 7. - Pontus von Brömssen, Aug 10 2024

More terms from Alois P. Heinz, Apr 05 2011

A375375 Inverse permutation to A375374.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 8, 10, 12, 9, 11, 15, 16, 13, 14, 17, 18, 21, 22, 25, 26, 19, 20, 23, 24, 31, 32, 33, 34, 27, 28, 29, 30, 35, 36, 37, 38, 43, 44, 45, 46, 51, 52, 53, 54, 39, 40, 41, 42, 47, 48, 49, 50, 63, 64, 65, 66, 67, 68, 69, 70, 55, 56, 57, 58, 59, 60

Offset: 1

Pontus von Brömssen, Aug 13 2024

nonn

Index entries for sequences that are permutations of the natural numbers.

3rd row of A375377.

Cf. A081241, A256231, A375374 (inverse permutation).

a(n) = A256231(A081241(n)).

cp's OEIS Frontend

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

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A185969 Let S be the sequence of power towers built of 2 and 3 sorted by their height and for equal heights - in lexicographic order: 2, 3, 2^2, 2^3, 3^2, 3^3, 2^2^2, 2^2^3 etc. A(n) = the permutation of indexes which reorders S by magnitude.

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A375375 Inverse permutation to A375374.

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