 Solution

The transformation $N=k/p$ is strictly decreasing for $p>0$. Therefore an interval $[L,U]$ with $0<L<U\le 1$ transforms toThe event $L\le p\le U$ is exactly the event $k/U\le N\le k/L$, so the approximate coverage is inherited. If the lower endpoint for $p$ reaches zero, the transformed upper endpoint for $N$ is $+\infty$; if the upper endpoint reaches one, the lower endpoint is $k$. Finally intersect with integer values $N>k$, rounding the lower bound upward and the upper bound downward. This respects the original discrete population constraint.