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:
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.
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 360110≤k≤1392490 but in the next 2000 numbers there are five which are above 20.
More theory is needed!