Fekete’s conjecture on repeated applications of the tangent function

Question 1

A high-school student named Erna Fekete made a conjecture to me via email three years ago,
which I could not answer. I’ve since lost touch with her.
I repeat her interesting conjecture here, in case anyone can provide updated
information on it.

Here is how she phrased it. Let $b(0) = 1$ and $b(n)= \tan( b(n-1) )$ .
In other words, $b(n)$ is the repeated application of $\tan(\;)$ to 1:
$\tan(1) = 1.56, \; \tan(\tan(1)) = 74.7, \; \tan^3(1) = -0.9, \; \ldots$

Let $a(n) = \lfloor b(n) \rfloor$ .
Her conjecture is:

Every integer eventually appears in the $a(n)$ sequence.

This sequence is not unknown; it
is A000319 in Sloane’s integer sequences.
Essentially hers is a question about the orbit of 1 under repeated $\tan(\;)$ -applications.
Her and my investigations at the time led us to believe it was an open problem.

Question 2

I had made the same conjecture as Fekete, apparently around the same time — mid-2007. In 2008 I verified that the first twenty million terms do not include 319. (I actually pushed the verification further, but I can’t find the more recent records at the moment.)

Because $\tan(x) - x = x^3/3 + O(x^5)$ , the function spends a lot of its time in a small neighborhood around $0$ . It escapes when it nears $\pi/2$ and quickly returns for many iterations.

A mostly-unexplained phenomenon presumably related to the above: there are long spans of small numbers followed by short, ‘productive’ spans with large numbers. $\tan^k(1)$ is “below 20 or so” (according to a 2008 email I sent) for $360110\le k\le1392490$ but in the next 2000 numbers there are five which are above 20.

More theory is needed!

Fekete’s conjecture on repeated applications of the tangent function

Answer

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