There are [divisibility sequences](https://en.wikipedia.org/wiki/Divisibility_sequence) of the form $\frac{p^k-1}{p-1}$ which have the property that if $a \mid b$, then $f(a) \mid f(b)$, ensuring (among other things) that only prime indices of the sequence can possibly be prime. This form grows as $2^n$ with $p=2$.

You can find significantly slower growth with the Fibonacci sequence, where the $n$th term is $ \frac{\varphi^n-\psi^n}{\sqrt 5} \approx 1.6^n$.

The slowest-growing form I know of is the recurrence relation defined by $S\_n = S\_{n-1} -2 S\_{n-2}$ with $S\_0 = 0, S\_1=1$. I don't know the closed-form expression offhand, but I think it grows as something closer to $1.4^n$.

So my question is, are there known slower-growing non-trivial divisibility sequences? By trivial, I'm referring to $\mathbb{N}$ or similar. I would guess that you should be able to generate one with an exponent arbitrarily close to $1$, but that's just speculation. Better yet, of course, would be one with sub-exponential growth, but I wouldn't be surprised if that were impossible.

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Loosely related question: it seems like every prime factor $a$ appearing in $\frac{p^k-1}{p-1}$ repeats with a period of $a-1$ (or one of its divisors) as $k$ varies, while every prime factor in Fibonacci-like recurrence relations repeats with a period of $a-1$, $a$, or $a+1$ (or divisors). Are there divisibility sequences where the factors have periods of e.g. $a-2$ or $a+2$? That is, anything other than $a$ or $a\pm 1$?