A [recent question][1] and [another question][2] were both marked as duplicates of an  [older one][3] which "has an answer." 

That is not off base and the answer is very nice but is described as a long comment. It concerns asymptotic growth rate of an integer function. Here I ask a different question regarding specific records.

The question addressed in the others is this:

For each positive integer $a$ and $b \lt a$ define a  decreasing sequence of positive integers by setting $b_0=b$ and $b_{i+1}=a \bmod b_i.$ Define $P(a,b)$ to be the length of this sequence (stopped when $b_n \vert a$ which would make $b_{n+1}=0.$)

The previous question is: **Bound $\max{P(a,b)}$ in terms of $a.$**

It is known that $P(a,b) = O(a^{1/3})$ and that infinitely often $P(a,b) > c \log a.$ The answer on the old question is offered as a long comment and gives a heuristic that $O(\log a)$ is the right order.

My question is:

> - What are the known records for the smallest $a$ such that $P(a,b)=n$ for some $b?$
- What is known about the records in general.

In the OEIS are sequences [A6538][4] for $a$ (it starts $1, 3, 5, 11, 11, 19$ ) and [A6537][5] for $b.$ (it starts $1, 2, 3, 4, 7, 12$) giving the smallest $a$ and for each the smallest $b$ with $P(a,b)=n$ so the fifth entries note that for $b=11$ one has $7 \rightarrow 4 \rightarrow 3 \rightarrow 2 \rightarrow 1.$ It turns out that no smaller $b$ gives a chain of length $5$ or even length $4$ which is why the $11$ is also in position four.

With the links in the entries the two lists are given up to $57$ with a comment from 2014 that $a=58017959$ suffices for length $58$ although perhaps some $10616759   \lt a \lt 58017959$ also gives that length.

So:
> Are there now any more known record $a$ values or good but not provably minimum values? What features do the pairs $(a,b)$ have? 

The more recent questions requested $a$ be prime.  It doesn't seem helpful to restrict in that way although is does assure that the sequence ends $b_n=1.$ Many of the records, but not all, are prime. for example $58017959=523 \cdot 110933.$



 


  [1]: https://mathoverflow.net/questions/306301/how-long-iterations-of-x-to-p-mod-x-can-be
  [2]: https://mathoverflow.net/questions/292284/consider-a-sequence-a-i-p-mod-a-i-1-where-p-is-a-prime-number-how-to-es
  [3]: https://mathoverflow.net/questions/164129/improving-known-bounds-for-pierce-expansions-cash-prize
  [4]: https://oeis.org/A006538
  [5]: https://oeis.org/A006537