This will be my second question here in math.stackexchange (so far).
This time, I am trying to consider the cases which give $\sigma(n) \equiv 0 \pmod 4$ whenever $n$ is odd. I get the following "lemma":
$\mathbf{Lemma:}$ If $n = \displaystyle\prod_{i = 1}^{r}{{p_i}^{\alpha_i}}$ is odd, then $\sigma(n) \equiv 0 \pmod 4$ when:
(i) there exists an $i$ such that the prime $p_i \equiv 3 \pmod 4$ has corresponding $\mathbf{odd}$ exponent $\alpha_i$ with ${p_i}^{\alpha_i}||n$; or
(ii) there exists an $i$ such that the prime $p_i \equiv 1 \pmod 4$ has corresponding exponent $\alpha_i \equiv 3 \pmod 4$ with ${p_i}^{\alpha_i}||n$; or
(iii) there exist $i$ and $j$ (with $i \neq j$) such that the primes $p_i$ and $p_j$ satisfy $p_i \equiv p_j \equiv 1 \pmod 4$ and have corresponding exponents $\alpha_i$ and $\alpha_j$ satisfying $\alpha_i \equiv \alpha_j \equiv 1 \pmod 4$ with ${p_i}^{\alpha_i}||n$ and ${p_j}^{\alpha_j}||n$.
My question now is: Is this (already) an exhaustive list of conditions for $\sigma(n) \equiv 0 \pmod 4$, whenever $n$ is odd?
Any ideas/comments/suggestions are most welcome.