 First successful Grover iterate

For one marked element among $N\ge 2$ candidates, set $\theta =\arcsin (N^{-1/2})$. The first nonnegative Grover search algorithm iteration count attaining marked probability at least ${\cos }^2\theta$ is the displayed integer. Before the first success interval, the angle $(2k+1)\theta$ is below $\pi /2-\theta$; entering that interval requires $k\ge \pi /(4\theta )-1$. Rounding up gives an angle less than $\pi /2+\theta$, so it cannot skip the interval. Moreover $k_{\min }<\pi /(4\theta )<\pi \sqrt{N}/4$. For $N=1$, the known candidate is already the answer and no iteration is needed; the two-dimensional marked--unmarked plane degenerates.