For a non-negative integer $\ell$, define the arithmetic functions $\sigma_{\ell}(k) = \sum_{d \mid k} d^{\ell}$ and $\bar{\sigma}_{\ell}(k) = \sum_{k = dd^{\prime}} (-1)^{(d^{\prime} - 1)/2} d^{\ell}$.
The function $\sigma(k) = \sigma_{1}(k)$ is the divisor-sum function.
Question: Let $k$ be an odd integer. Is the following congruence known?
\begin{align*} 3\sigma(k) &\equiv \sigma_{3}(k) + 2 \bar{\sigma}_{0}(k) &\pmod{16} \end{align*}
Thanks!
Update: What about this one? \begin{align*} \bar{\sigma}_{2}(k) &\equiv \bar{\sigma}_{0}(k) &\pmod{4} \end{align*}