Fekete’s conjecture on repeated applications of the tangent function

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(n1)).
In other words, b(n) is the repeated application of tan() to 1:
tan(1)=1.56,tan(tan(1))=74.7,tan3(1)=0.9,

Let a(n)=b(n).
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.

Answer

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=x3/3+O(x5), the function spends a lot of its time in a small neighborhood around 0. It escapes when it nears π/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. tank(1) is “below 20 or so” (according to a 2008 email I sent) for 360110k1392490 but in the next 2000 numbers there are five which are above 20.

More theory is needed!

Attribution
Source : Link , Question Author : Joseph O’Rourke , Answer Author : Martin Sleziak

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