Sample size for a weak correlation to be significant

You have $n$ data pairs $(x_i,y_i)$ with both $x$ and $y$ centred to mean zero, and you regress $y$ on $x$ through the origin, $y=\hat\beta x+\varepsilon$. The sample correlation is $\tfrac{1}{10\sqrt3}$. What is the smallest $n$ for which the one-sided test of $H_0:\beta=0$ against $\beta>0$ is significant at the $5\%$ level? Use $1.645$ and $1.96$ as the $95$th and $97.5$th percentiles.

$811$. For a no-intercept fit on centred data with $n-1$ residual degrees of freedom, the slope $t$ statistic simplifies to $t=r\sqrt{n-1}/\sqrt{1-r^2}$. With $r^2=\tfrac1{300}$ this is $\sqrt{(n-1)/299}$, which first exceeds $1.645$ when $n-1=810$.