Is there an equation that relates the parity of $\sigma_0$ (the number of divisors) to the parity of $\sigma_1$ (the sum of divisors)?
Added February 21 2017
From S.C.B.'s comment: "$\sigma_{0}(n)$ is odd iff $n$ is a square. $\sigma_{1}(n)$ is odd iff $n$ is a square or two times a square." Therefore, $\sigma_{0}(n)$ and $\sigma_{1}(n)$ are both odd when $n$ is a square.
I would still be interested in receiving answers to the following question:
Earlier Version of Question
Is there an equation that relates $\sigma_0$ (the number of divisors) to $\sigma_1$ (the sum of divisors)?