Let $\omega(m)$ be the number of distinct prime divisors of a positive integer $m>1$. I am interested in the relative orders in which the numbers $\omega(n+1),...,\omega(n+k)$ can occur.
Given a positive integer $k\geq 2$ and a permutation $\sigma$ of $\{1,2,...,k\}$, are there infinitely many positive integers $n$ such that $$\omega(n+\sigma(1))<\omega(n+\sigma(2))<...<\omega(n+\sigma(k))?\quad \quad \quad (1)$$
The result should of course be true as there is no reason to expect correlation between the prime factorizations of consecutive numbers, but I do not know how to prove this. An easy remark is that if one can find for any $k$ and $\sigma$ one $n$ with the desired property, then there are infinitely many $n$ satisfying the claim. An obvious generalization is
Given $k$ and a permutation $\sigma$ of $\{1,2,...,k\}$, is the natural density of those $n$ that satisfy $(1)$ equal to $\frac{1}{k!}$?
This seems very likely since the Erdös-Kac theorem says that $\frac{\omega(n)-\log \log n}{\sqrt{\log \log n}}$ is distributed according to the standard normal distribution. If the random variables $\frac{\omega(n+i)-\log \log n}{\sqrt{\log \log n}}$ were independent even in some weak sense, question 1 would be resolved.
An affirmative answer to my first question would follow from Dickson's conjecture, but my guess is that the answer is known unconditionally, and Dickson's conjecture is a very strong assumption anyway.
I would also like to know whether there is an elementary proof of the special case of question 1 that the sequence $\{\omega(n)\}_{n=2}^{\infty}$ contains arbitrarily long strictly increasing subsequences consisting of consecutive terms.
