$$\\ N_1 = \Bigg\{ z_{1-\alpha/2}*\sqrt{\bar{p}*\bar{q}*(1+\frac{1}{k}}) + z_{1-\beta}*\sqrt{p_1*q_1+(\frac{p_2*q_2}{k}}) \Bigg\}^2/\Delta^2\\q_1 = 1-p_1\\q_2 = 1-p_2\\\bar{p} = \frac{p_1+kp_2}{1+K}\\\bar{q} = 1-\bar{p}\\ N_1 = \Bigg\{ 1.96*\sqrt{0.315*0.685*(1+\frac{1}{1}}) + 0.84*\sqrt{0.35*0.65+(\frac{0.28*0.72}{1}}) \Bigg\}^2/0.07^2\\ N_1 = 690\\ N_2 = K*N_1 = 690$$
p_{1}, p_{2} = proportion (incidence) of groups #1 and #2
Δ = |p_{2}-p_{1}| = absolute difference between two proportions
n_{1} = sample size for group #1
n_{2} = sample size for group #2
α = probability of type I error (usually 0.05)
β = probability of type II error (usually 0.2)
z = critical Z value for a given α or β
K = ratio of sample size for group #2 to group #1