Merca's conjectures on odd divisor-function convolutions

Let $P_m(k)$ be the generalized $m$-gonal number

$$P_m(k):=\left(\frac{m}{2}-1\right)k^2-\left(\frac{m}{2}-2\right)k,$$

and let

$$\sigma_{\mathrm{odd}}(n):=\sum_{\substack{d\mid n\d\ \mathrm{odd}}}d,$$

with $\sigma_{\mathrm{odd}}(n):=0$ for $n\leq 0$. For positive integers $m$ and $n$, consider the three congruences

$$\sum_{k=-\infty}^{\infty}\sigma_{\mathrm{odd}}(n-P_m(k))\equiv \begin{cases} n \allowbreak \if@display\mkern18mu\else\mkern8mu\fi \pmodx 2&\text{if }n=P_m(j),\ j\in\mathbb{Z},\\ 0 \allowbreak \if@display\mkern18mu\else\mkern8mu\fi \pmodx 2&\text{otherwise}, \end{cases}$$ $$\sum_{k=-\infty}^{\infty}\sigma_{\mathrm{odd}}(n-P_5(k))\equiv \begin{cases} n \allowbreak \if@display\mkern18mu\else\mkern8mu\fi \pmodx m&\text{if }n=P_5(j),\ j\in\mathbb{Z},\\ 0 \allowbreak \if@display\mkern18mu\else\mkern8mu\fi \pmodx m&\text{otherwise}, \end{cases}$$

and

$$\sum_{k=-\infty}^{\infty}(-1)^{P_3(-k)}\sigma_{\mathrm{odd}}(n-P_5(k))\equiv \begin{cases} (-1)^{P_3(-j)}n \allowbreak \if@display\mkern18mu\else\mkern8mu\fi \pmodx m&\text{if }n=P_5(j),\ j\in\mathbb{Z},\\ 0 \allowbreak \if@display\mkern18mu\else\mkern8mu\fi \pmodx m&\text{otherwise}. \end{cases}$$

Merca's conjectures. The first congruence holds for all $n\in\mathbb{Z}^+$ if and only if $m\in\{5,6\}$; the second holds for all $n\in\mathbb{Z}^+$ if and only if $m\in\{2,3,6\}$; and the third holds for all $n\in\mathbb{Z}^+$ if and only if $m\in\{2,4\}$. The paper proves these assertions, so they are now theorems rather than open conjectures.