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A299235 Number of 2's in the n-th {2,3}-power tower; see Comments. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Suppose that S is a set of real numbers.  As 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-1)); (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.

LINKS

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

EXAMPLE

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

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

Cf. A299229, A299236 (complement).

Sequence in context: A230079 A105400 A194516 * A245840 A033774 A033804

Adjacent sequences:  A299232 A299233 A299234 * A299236 A299237 A299238

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Feb 06 2018

STATUS

approved

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Last modified August 7 15:30 EDT 2020. Contains 336276 sequences. (Running on oeis4.)