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Your $a_n$ is a divisibility sequence, right? That means $a_n \mid a_{mn}$, so to get prime values, you'll pretty much need the index to be prime. Next, note that $a_n$ is approximately $c^n$. So the …

Rather than defining a sequence in which the recursion depends on the number of iterations, it might be advantageous to simply consider this as iteration of the $2$-dimensional polynomial recursion $$ …

I think most (all?) of this follows from basic properties of linear recurrences. The characteristic polynomial of your $p$-step Fibonacci numbers is $x^p-x^{p-1}-x^{p-2}-\cdots-x-1$. You seem to have …

I'm not quite sure what you mean by an "analytic formula." As Fedor Petrov indicated, there is unlikely to be a closed formula. However, there is a convergent power series. More precisely, consider it …

This is a second answer with a somewhat different viewpoint from my other answer. It is an expansion, in some sense, of Gerry Myerson's answer. There is a general theory for estimating these sorts of …

Actually, the binary linear recurrence case is pretty precise, especially if $p\ge3$ and you're working over $\mathbb Q$, and not over a field where $p$ is ramified. Let $r(p)$ denote the rank of appa …