Find the sum of positive integers $n$ less than $2021$ such that $n^{3 \cdot 7 \cdot 23} \equiv -1 \pmod{2021}$.
I was making an elementary number theory problem using the year number $2021=43 \cdot 47$. Of course the number of solution is $21 \times 23= 483$. By the way, I have calculated the sum of all solutions to get $493124$ which is exactly $2021 \cdot (483+5)/2$. The exponent $3 \cdot 7 \cdot 23$ is $\varphi(2021)/4$. I wonder if there could be a systemetic explanation about the result. Thanks for your attention.
PS: After some test on the other numbers, I guess the number $483+5$ is fairly random consquence of the distribution of the solution. Therefore, any systemetic explanation without counting all the solutions on some interval, if any, likely to be very hard.