An equation concerning perfect numbers

Let $p_n=\max\{p_1,...,p_n\}$.

Thus, $1+p_i$ for some $i$ is divisible by $p_n$, which is possible only for $n=2$, $p_2=3$ and $p_1=2$.

Since $1+p_i$ is divisible by $p_n$, we see that $1+p_i\geq p_n$ and $p_n\neq p_i$, which gives $p_n\geq1+p_i$.

Thus, $p_n=1+p_i,$ which is possible, when $p_i=2$ and $p_n=3$.