Problem 4

For $n=5$ and $n=17$ (both $\equiv5\pmod{12}$, below the threshold $n\ge29$ of step 6), a direct check of all pairs $(a,b)$ with $a\equiv b\pmod3$ and $2(a+b)-3\in\{5,17\}$ shows only $(2,2)$, $(2,8)$, and $(5,5)$ (up to swapping $a,b$), none of which is coprime. So $n=5$ and $n=17$ are not achievable, while every other $n\equiv1,5\pmod6$ is realized by step 6. Hence the possible values of $n$ are exactly the integers with $n\equiv1\pmod6$ or $n\equiv5\pmod6$, excluding $n=5$ and $n=17$.