It works for: $n = \{-507023, -170349, -103015, -58127, -35685, -22223,$ $-13255, -9581, -8781, -4623, -4242, -3685, -3149, -3015, -1005,$ $-871, -335, 222, 603, 4761, 5561, 9235, 18203, 31665, 54107, $ $98995, 166329, 503003\}$.

We use difference of squares to get $n(n+2010)=(n+1005)^2-1005^2=k^2 \implies (n+1005+k)(n+1005-k)=1005^2=3^2 \cdot 5^2 \cdot 67^2$