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.

EXAMPLE

Northwest corner:

1 4 5 11 12 15

3 8 9 10 18 20

6 14 17 19 21 30

13 28 29 35 39 43

27 56 57 59 71 79

55 112 113 115 119 143

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 = 400; g[k_] := If[EvenQ[k], {2}, {3}];

f = 6; While[f < 13, n = f; While[n < z, p = 1;

While[p < 18, 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_] := Select[Range[5000], Count[t[#], 2] == n &]

TableForm[Table[r[n], {n, 1, 15}]] (* this array *)

w[n_, k_] := r[n][[k]];

Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* this sequence *)

CROSSREFS

KEYWORD

AUTHOR

Clark Kimberling, Feb 08 2018

STATUS

approved