A299326 - OEIS

%I #14 Aug 07 2024 15:21:55

%S 2,5,7,8,12,16,11,18,26,34,14,24,38,54,70,20,30,50,78,110,142,22,42,

%T 62,102,158,222,286,28,46,86,126,206,318,446,574,32,58,94,174,254,414,

%U 638,894,1150,36,66,118,190,350,510,830,1278,1790,2302

%N 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.

%C 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.

%C As sequences, this one and A299325 partition the positive integers.

%D 1

%e Northwest corner:

%e 2     5    8   11   14   20   22

%e 7    12   18   24   30   42   46

%e 16   26   38   50   62   86   94

%e 34   54   78  102  126  174  190

%e 70  110  158  206  254  350  382

%t t[1] = {2}; t[2] = {3}; t[3] = {2, 2}; t[4] = {2, 3}; t[5] = {3, 2};

%t t[6] = {2, 2, 2}; t[7] = {3, 3};

%t t[8] = {3, 2, 2}; t[9] = {2, 2, 3}; t[10] = {2, 3, 2};

%t t[11] = {3, 2, 3}; t[12] = {3, 3, 2};

%t z = 500; g[k_] := If[EvenQ[k], {2}, {3}];

%t f = 6; While[f < 13, n = f; While[n < z, p = 1;

%t    While[p < 17, m = 2 n + 1; v = t[n]; k = 0;

%t    While[k < 2^p, t[m + k] = Join[g[k], t[n + Floor[k/2]]]; k = k + 1];

%t    p = p + 1; n = m]]; f = f + 1]

%t s = Select[Range[60000], Count[First[Split[t[#]]], 2] == 0 & ];

%t r[n_] := Select[s, Length[First[Split[t[#]]]] == n &, 12]

%t TableForm[Table[r[n], {n, 1, 10}]]  (* this array *)

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

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

%Y Cf. A299229, A299325.

%K nonn,easy,tabl

%O 1,1

%A _Clark Kimberling_, Feb 08 2018