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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 >=0; 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 (list; table; graph; refs; listen; history; text; internal format)
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-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 A299325 partition the positive integers.

REFERENCES

1

LINKS

Table of n, a(n) for n=1..55.

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}]]  (* A299326, array *)

w[n_, k_] := r[n][[k]];

Table[w[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* A299326, sequence *)

CROSSREFS

Cf. A299229, A299325.

Sequence in context: A192111 A294635 A032402 * A280848 A190900 A216572

Adjacent sequences:  A299323 A299324 A299325 * A299327 A299328 A299329

KEYWORD

nonn,easy,tabl

AUTHOR

Clark Kimberling, Feb 08 2018

STATUS

approved

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Last modified May 27 02:54 EDT 2019. Contains 323597 sequences. (Running on oeis4.)