Assume for some $t>1$ we had $a_t<a_1$. From $a_t + a_m = a_{tm} + a_1$ would follow $a_m > a_{tm}$ for all positive integers $m$, thus $a_m > a_{tm} > a_{t^2m} > \cdots$. But this infinite descent of positive integers is impossible. So $a_t\geq a_1$ for all $t\geq 1$. From $a_p + a_{q/p} = a_{q} + a_1$ and $a_{q/p} \geq a_1$ then follows $a_p \leq a_{q}$ whenever $p\mid q$.