Marinada '21

[lang=hr] Za $n \in \mathbb{N}$ definirajmo skupove $M_n$ na sljedeći način: \\ - $M_1 = \{ 2 \}$ \\ - za $n \geq 2$, $M_n$ je najmanji skup koji sadrži $M_{n-1}$ te koji sadrži sve proste faktore od $p_1p_2...p_k+1$, za sve podskupove $ \{p_1,p_2,...,p_k\}$ od $M_{n-1}$\\ Definirajmo $M = M_1 \cup M_2 \cup M_3 \cup ...$, te neka je $m(n)$ oznaka za $n$-ti po redu broj u $M$, a $p(n)$ za $n$-ti po redu prost broj. \\ Odredi $m(202142069)-p(202142069)$. [/lang] [lang=en] For $n \in \mathbb{N}$ define the set $M_n$ as follows: \\ - $M_1 = \{ 2 \}$ \\ - for $n \geq 2$, $M_n$ is the smallest set that contains $M_{n-1}$ and that also contains all of the factors of $p_1p_2...p_k + 1$ for all subsets $\{p_1, p_2, ..., p_k\}$ of $M_{n-1}$. \\ Define $M = M_1 \cup M_2 \cup M_3 \cup ...$ and let $m(n)$ denote the $n$-th number in the set $M$ (when ordered by value), and let $p(n)$ denote the $n$-th prime. \\ Calculate $m(202142069)-p(202142069)$. [/lang]