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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
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
1,1
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.
REFERENCES
1
EXAMPLE
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
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 *)
CROSSREFS
Sequence in context: A294635 A362591 A032402 * A280848 A190900 A216572
KEYWORD
nonn,easy,tabl
AUTHOR
Clark Kimberling, Feb 08 2018
STATUS
approved