OFFSET
1,7
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 2's. The position of the n-th 0 in the sequence is the rank of the n-th {2}-power tower, given by 7*2^(n-3) - 1 for n > 2.
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..10000
EXAMPLE
t(13) = (2,2,2,2), so that a(13) = 0.
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 06 2018
STATUS
approved