Marinada '22

[lang=hr] Za svaki neparan prost broj $p$, definiramo rekurzivan niz $(a_n(p))_{n \geq 0}$ tako da je $a_0(p)=0$ i $$a_{n+1}(p)=(q+1)\left((a_n(p))^{q+1}+(a_n(p))^{q}+a_n(p)+1 \right)$$ za $n \geq 0$, gdje je $q=\frac{p-1}{2}$. Za prirodan broj $m \geq 2$ kažemo da je divan ako postoji prost $p$ takav da $p \mid a_m(p)$, te $p \nmid a_k(p)$ za svaki $k \in \{1,\ldots, m-1\}.$ Drugim riječima, $m$ je najmanji pozitivan indeks takav da $p \mid a_m(p)$. Koliko ima divnih brojeva manjih od $1000$? [/lang] [lang=en] For every odd prime $p$, define a recursive sequence $(a_n(p))_{n \geq 0}$ by $a_0(p)=0$ and $$a_{n+1}(p)=(q+1)\left((a_n(p))^{q+1}+(a_n(p))^{q}+a_n(p)+1 \right)$$ for $n \geq 0$, where $q=\frac{p-1}{2}$. We say a positive integer $m \geq 2$ is lovely if there exists a prime number $p$ such that $p \mid a_m(p)$, and $p \nmid a_k(p)$ for every $k \in \{1,\ldots, m-1\}.$ In other words, $m$ is the smallest positive index such that $p \mid a_m(p)$. How many lovely numbers less than $1000$ are there? [/lang]