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Conjecture 7. For any odd prime $p$, \begin{equation} \sum_{1\leq n \leq (p-1)p}a(n) \equiv \begin{cases} p \pmod{p^2} &\mbox{if $p \equiv 1 \pmod 4$ }\\ 0 \pmod{p^2} &\mbox{if $p \equiv 3 \pmod 4$ } \end{cases}. \end{equation} It is true for $3 \leq p \leq 19$.

Conjecture 7. For any odd prime $p$, \begin{equation} \sum_{n=1}^{p(p-1)}a(n) \equiv \begin{cases} p \pmod{p^2} &\mbox{if $p \equiv 1 \pmod 4$ }\\ 0 \pmod{p^2} &\mbox{if $p \equiv 3 \pmod 4$ } \end{cases}. \end{equation} It is true for $3 \leq p \leq 19$.

ADDED (2021-10-04)

Conjecture 8. For any prime $p$, $$\sum_{n=1}^{p(p-1)}\left(\frac{a(n)}{p}\right)\equiv 0 \pmod p,$$ where $\left(\frac{\cdot}p\right)$ is the Legendre symbol.

It is true for $2 \leq p \leq 19$.

ADDED (2021-9-24)

ADDED (2021-9-27)