Let $\omega(n)$ refer to the number of distinct prime factors of $n$.
I'm curious about what values of $\frac{\omega(n-1)}{\omega(n)}$ (which I'll call $\alpha(n)$ for convenience) can be realized by composite numbers.
As it turns out, there are many composite $n$ such that $\alpha(n)$ is $2,3,4,5,6$ ...
$2$ is very easy to achieve with a prime power.
16
25
27
49
So is $3$.
121
169
256
343
And so on up to 6.
175561
212521
326041
434281
Seven has a few but they're harder to find.
2042041
7447441
9393931
I'm curious whether every possible for $\alpha(n)$ is realizable by some composite $n$.
The examples that have turned up so far are prime powers and mostly squares of primes at that.
Update #1:
Interestingly, if you remove the restriction that $n$ has to be composite, there are relatively small solutions for $\alpha^{-1}(8)$ and $\alpha^{-1}(9)$.
Here are the solutions for $\alpha^{-1}(9)$ that I've found so far, all are prime.
300690391
340510171
358888531
397687291
406816411
Another pattern that seems to hold is that the smallest solution to $\alpha^{-1}(n)$ is less than $10^n$.
Update #2
It's possible that $\omega(-1 + 2^{\mathbb{N}_{\ge 1}})$ is $\mathbb{N}_{\ge 1}$. Since powers of two have exactly one prime factor, this is useful for establishing the truth of the lemma.
This OEIS sequence has the number of distinct factors of $-1+2^n$ https://oeis.org/A046800.
Some cursory investigation using pari/gp also suggests that this hits every positive integer.
$$ \omega(-1 + 64 ^ {60}) = 29 \\ \omega(-1 + 64 ^ {66}) = 27 \\ \omega(-1 + 64 ^ {70}) = 34 $$