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Consider the nine consecutive natural numbers starting from $1584614377$.

n = 1584614377 no. of divisors: 2
n = 1584614378 no. of divisors: 4
n = 1584614379 no. of divisors: 8
n = 1584614380 no. of divisors: 12
n = 1584614381 no. of divisors: 14
n = 1584614382 no. of divisors: 16
n = 1584614383 no. of divisors: 32
n = 1584614384 no. of divisors: 40
n = 1584614385 no. of divisors: 48

Their number of divisors is strictly increasing. This is an examples of a increasing sequence of the number of divisors of length $9$. More A075028.

For every positive integer $n \ge 1$, we calculate far we can go before the sequence of increasing divisors breaks. For some $n$ the sequence will break immediately at $n+1$ while for others it might go longer. On an average how far we can go before the sequence breaks? Let $I_n$ be the average of length of increasing sequence for the first $n$ natural numbers. Experimental data shows that $I_{10^6} \approx 1.537$ and $I_{1.2 \times 10^{10}} \approx 1.5537$. It will be nice if it converges to well known constant such as $\pi/2$. A heuristic by Greg Martin mentioned in the comments of the linked MSE post suggests that the average length might be $e-1$.

Question 1: What is $\lim_{n \to \infty}I_n$?

Instead of an increasing sequence of divisors, let is consider a decreasing sequence of divisors and let $D_n$, be the average of length of decreasing sequence for the first $n$ natural numbers.

Question 2: Is it true that $\lim_{n \to \infty}I_n = \lim_{n \to \infty}D_n$?

Note: This question got upvotes in MSE but was unanswered. Hence posting in MO.

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    $\begingroup$ A comment on MSE gives a link to the corresponding OEIS page, which indicates an increasing sequence of length 12. $\endgroup$
    – Emil Jeřábek
    Commented Jan 20, 2022 at 10:22
  • $\begingroup$ How what? Did you check the link? $\endgroup$
    – Emil Jeřábek
    Commented Jan 20, 2022 at 15:31
  • $\begingroup$ @EmilJeřábek Yes but, I still don't see how the average length will be 12. $\endgroup$
    – Nilotpal Kanti Sinha
    Commented Jan 20, 2022 at 15:40
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    $\begingroup$ I never said anything about average length. I'm pointing out that even though you write "The longest such sequence I have found so far is the above sequence of length 9" in the question, someone has in fact given you a sequence of length 12 two months ago, hence the statement in the question is misleading. $\endgroup$
    – Emil Jeřábek
    Commented Jan 20, 2022 at 16:02
  • $\begingroup$ @EmilJeřábek OK got you now. Clarified. $\endgroup$
    – Nilotpal Kanti Sinha
    Commented Jan 20, 2022 at 16:04

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