= Solution
Let $z_p=\Phi^{-1}(p)$ denote a <quantile> of the <standard normal distribution>. Reject when $W_n>z_{1-\alpha}$. The <statistical power> at the specified positive effect is
$$
P_{\delta^*}(W_n>z_{1-\alpha})=1-\Phi\left(z_{1-\alpha}-\frac{\sqrt n\delta^*}{\sigma}\right).
$$
Equating this to $1-\beta$ and using $z_\beta=-z_{1-\beta}$ gives $\sqrt n\delta^*/\sigma=z_{1-\alpha}+z_{1-\beta}$. Thus, for the usual target $1-\beta>\alpha$,
$$
\boxed{n=\left\lceil\frac{\sigma^2}{(\delta^*)^2}\left(z_{1-\alpha}+z_{1-\beta}\right)^2\right\rceil.}
$$
Rounding up ensures at least the target <statistical power>. This <normal-mean sample size calculation> assumes a positive integer <sample size>; if a requested power is at most $\alpha$, every positive <sample size> already exceeds that target for $\delta^*>0$, and one should not square a negative <quantile> sum to impose an unnecessary lower bound.
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