A299325 Rectangular array by antidiagonals: row n gives the ranks of {2,3}-power towers that start with n 2's, for n >= 1; see Comments.

1, 4, 3, 10, 9, 6, 15, 21, 19, 13, 17, 31, 43, 39, 27, 23, 35, 63, 87, 79, 55, 25, 47, 71, 127, 175, 159, 111, 29, 51, 95, 143, 255, 351, 319, 223, 33, 59, 103, 191, 287, 511, 703, 639, 447, 37, 67, 119, 207, 383, 575, 1023, 1407, 1279, 895, 41, 75, 135, 239

Offset: 1

Author

Clark Kimberling, Feb 08 2018

Keywords

• nonn
• easy
• tabl

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.

As sequences, this one and A299326 partition the positive integers.

Examples

			Northwest corner:
   1    4    10    15    17    23    25
   3    9    21    31    35    47    51
   6   19    43    63    71    95   103
  13   39    87   127   143   191   207
  27   79   175   255   287   383   415
		

Crossrefs

Cf. A299229, A299326.

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 = 500; g[k_] := If[EvenQ[k], {2}, {3}];
f = 6; While[f < 13, n = f;  While[n < z, p = 1;
   While[p < 17, 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]
s = Select[Range[60000], Count[First[Split[t[#]]], 3] == 0 & ];
r[n_] := Select[s, Length[First[Split[t[#]]]] == n &, 12]
TableForm[Table[r[n], {n, 1, 11}]]  (* this array *)
w[n_, k_] := r[n][[k]];
Table[w[n - k + 1, k], {n, 11}, {k, n, 1, -1}] // Flatten (* this sequence *)