When does scaling an elliptic curve point align with the standard notion of scaling coordinate points?
Let $P = (x_0,y_0)$ be a point on an elliptic curve $E$ over a prime $p$. When can we find $n$ and $\lambda$ so that
$$nP = (\lambda x_0, \lambda y_0)$$
where $nP$ is normal elliptic curve multiplication? (We can span all the points for suitable $n$ and generator $P$ but I'm looking for specific constructions of for $n$)