A299240 Ranks of {2,3}-power towers in which #2's > #3's; see Comments.

1, 3, 6, 8, 9, 10, 13, 14, 17, 19, 21, 27, 28, 29, 30, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 51, 55, 56, 57, 58, 59, 60, 61, 63, 71, 72, 73, 75, 79, 80, 81, 83, 87, 88, 89, 91, 95, 103, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124

Offset: 1

Author

Clark Kimberling, Feb 07 2018

Keywords

• nonn
• easy

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.

This sequence together with A299241 and A299242 partition the positive integers.

Examples

			The first six terms are the ranks of these towers: t(1) = (2), t(3) = (2,2), t(6) = (2,2,2), t(8) = (3,2,2), t(9) = (2,2,3), t(10) = (2,3,2).
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A299229, A299241, A299242.

Programs

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 = 190; g[k_] := If[EvenQ[k], {2}, {3}]; f = 6;
While[f < 13, n = f; While[n < z, p = 1;
  While[p < 12, 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]
Select[Range[1000], Count[t[#], 2] > Count[t[#], 3] &];   (* A299240 *)
Select[Range[1000], Count[t[#], 2] == Count[t[#], 3] &];  (* A299241 *)
Select[Range[1000], Count[t[#], 2] < Count[t[#], 3] &];   (* A299242 *)