6. (i) Find the smallest positive integer $a$ such that $\sigma(n)=a$ has no solution, exactly one solution, exactly two solutions, and exactly three solutions;
(ii) There exist infinitely many $a$ for which $\sigma(n)=a$ has no solution.
6. (i) For $a=2,1,12,24$, the corresponding solutions are: $\sigma(1)=1, \sigma(6)=\sigma(11)=12$; $\sigma(14)=\sigma(15)=\sigma(23)=24$. To utilize the following conclusion: when $r \geqslant 3$, $\sigma\left(p_{1}^{\circ} p_{2}^{\alpha_{2}^{2}} p_{3}^{\alpha_{3}}\right) \geqslant 72$, and when $r \leqslant 2$ there are only the above solutions; (ii) when $a=3^{k}(k \geqslant 2)$, $\sigma(n)=a$ has no solutions.