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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.
2
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
1,2
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.
As sequences, this one and A299326 partition the positive integers.
EXAMPLE
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
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 *)
CROSSREFS
Sequence in context: A213768 A325997 A075563 * A316196 A081617 A103252
KEYWORD
nonn,easy,tabl
AUTHOR
Clark Kimberling, Feb 08 2018
STATUS
approved