A299242 - OEIS

A299242

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

2, 7, 11, 12, 15, 16, 24, 26, 32, 33, 34, 38, 42, 46, 48, 49, 50, 52, 53, 54, 64, 65, 66, 67, 68, 69, 70, 78, 86, 94, 98, 100, 101, 102, 106, 108, 109, 110, 126, 130, 132, 133, 134, 136, 137, 138, 139, 140, 141, 142, 150, 154, 156, 157, 158, 166, 170, 172

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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))); (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 A299240 and A299241 partition the positive integers.

LINKS

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

EXAMPLE

The first six terms are the ranks of these towers: t(2) = (3), t(7) = (3,3), t(11) = (3,2,3), t(12) = (3,3,2), t(15) = (2,3,3), t(16) = (3,3,3).

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;