A recent question and another question were both marked as duplicates of an older one 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 for $a$ (it starts $1, 3, 5, 11, 11, 19$ ) and A6537 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 $58017959=523 \cdot 110933.$

uniquely interesting? I didn't think so. $\endgroup$ – Aaron Meyerowitz Jul 23 '18 at 21:35