OMCPLP 2021 Problem 6

A positive integer $n$ is called $omopeiro$ if there exists $n$ non-zero integers that are not necessarily distinct such that $2021$ is the sum of the squares of those $n$ integers. For example, the number $2$ is not an $omopeiro$ , because $2021$ is not a sum of two non-zero squares, but $2021$ is an $omopeiro$ , because $2021=1^2+1^2+ \dots +1^2$ , which is a sum of $2021$ squares of the number $1$ .

Prove that there exist more than 1500 $omopeiro$ numbers.

Note: proving that there exist at least 500 $omopeiro$ numbers is worth 2 points.