A299326 Rectangular array by antidiagonals: row n gives the ranks of {2,3}-power towers that start with n 3's, for n >= 1; see Comments.

2, 5, 7, 8, 12, 16, 11, 18, 26, 34, 14, 24, 38, 54, 70, 20, 30, 50, 78, 110, 142, 22, 42, 62, 102, 158, 222, 286, 28, 46, 86, 126, 206, 318, 446, 574, 32, 58, 94, 174, 254, 414, 638, 894, 1150, 36, 66, 118, 190, 350, 510, 830, 1278, 1790, 2302

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 A299325 partition the positive integers.

Examples

			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
		

References

• 1

Crossrefs

Cf. A299229, A299325.

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[#]]], 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 *)