A299235 Number of 2's in the n-th {2,3}-power tower; see Comments.

1, 0, 2, 1, 1, 3, 0, 2, 2, 2, 1, 1, 4, 3, 1, 0, 3, 2, 3, 2, 3, 2, 2, 1, 2, 1, 5, 4, 4, 3, 2, 1, 1, 0, 4, 3, 3, 2, 4, 3, 3, 2, 4, 3, 3, 2, 3, 2, 2, 1, 3, 2, 2, 1, 6, 5, 5, 4, 5, 4, 4, 3, 3, 2, 2, 1, 2, 1, 1, 0, 5, 4, 4, 3, 4, 3, 3, 2, 5, 4, 4, 3, 4, 3, 3, 2

Offset: 1

Author

Clark Kimberling, Feb 06 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.

Every nonnegative integer occurs infinitely many times in the sequence. In particular, a(n) = 0 when the tower consists exclusively of 3's. The position of the n-th 0 in the sequence is the rank of the n-th {3}-power tower, given by 9*2^(n-2)-2 for n > 1.

Examples

			t(80) = (3,2,2,2,2,3), so that a(80) = 4.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A299229, A299236 (complement).

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]
Table[Count[t[n], 2], {n, 1, 100}];  (* A299235 *)
Table[Count[t[n], 3], {n, 1, 100}];  (* A299236 *)