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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

Table of n, a(n) for n=1..59.

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}]]  (* 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: A147756 A213768 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 April 18 08:37 EDT 2019. Contains 322209 sequences. (Running on oeis4.)