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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 >=0; see Comments. 3
 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 (list; table; graph; refs; listen; history; text; internal format)
 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-1)); (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. LINKS 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 = {2}; t = {3}; t = {2, 2}; t = {2, 3}; t = {3, 2}; t = {2, 2, 2}; t = {3, 3}; t = {3, 2, 2}; t = {2, 2, 3}; t = {2, 3, 2}; t = {3, 2, 3}; t = {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, Count[First[Split[t[#]]], 3] == 0 & ]; r[n_] := Select[s, Length[First[Split[t[#]]]] == n &, 12] TableForm[Table[r[n], {n, 1, 11}]]  (* A299325, array *) w[n_, k_] := r[n][[k]]; Table[w[n - k + 1, k], {n, 11}, {k, n, 1, -1}] // Flatten (*   A299325, sequence *) CROSSREFS Cf. A299229, A299326. Sequence in context: A213768 A325997 A075563 * A316196 A081617 A103252 Adjacent sequences:  A299322 A299323 A299324 * A299326 A299327 A299328 KEYWORD nonn,easy,tabl AUTHOR Clark Kimberling, Feb 08 2018 STATUS approved

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Last modified August 17 15:35 EDT 2022. Contains 356189 sequences. (Running on oeis4.)