σ is an odd integer if and only if n is a perfecdt square or twice a perfect square.
[Hint: If p i s an odd prime, then 1+p+p^2+...+p^k is odd only when k is even]
It is a basic fact that if $\displaystyle n=p_1^{a_1}\cdot \ldots \cdot p_k^{a_k}\in\mathbb{N}$ is the prime decomposition of a natural number, then $\displaystyle \sigma(n)=\prod\limits_{i=1}^k\frac{p_i^{a_i+1}-1}{p_i-1}=\prod\limits_{i=1}^k\left(\sum\limits_{m=0}^{a_ i}p_i^m\right)$
Well, now just use the hint...
Tonio