1,1

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 A299325 partition the positive integers.

1

Northwest corner:

2 5 8 11 14 20 22

7 12 18 24 30 42 46

16 26 38 50 62 86 94

34 54 78 102 126 174 190

70 110 158 206 254 350 382

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[#]]], 2] == 0 & ];

r[n_] := Select[s, Length[First[Split[t[#]]]] == n &, 12]

TableForm[Table[r[n], {n, 1, 10}]] (* this array *)

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

Table[w[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* this sequence *)

Cf. A299229, A299325.

Sequence in context: A294635 A362591 A032402 * A280848 A190900 A216572

Adjacent sequences: A299323 A299324 A299325 * A299327 A299328 A299329

nonn,easy,tabl

Clark Kimberling, Feb 08 2018

approved