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Two optimal frameworks for multichannel spectrum sensing in cognitive radio networks are presented. Both frameworks search for multiple secondary transmission opportunities over a number of narrowband channels, enhancing the secondary network performance while respecting the primary network integrity and keeping the interference limited. Considering a sequential periodic sensing scheme with non-uniform channel sensing durations, the original sensing problems are formulated as optimization problems to maximize the throughput capacity of the secondary network subject to some bounds/limits on the interference to the primary network. The first framework, referred to as sequential multichannel joint detection, considers the aggregate (weighted) interference on the primary network as the constraint. The second framework, known as decoupled sequential multichannel joint detection, assumes that interference is limited in each channel separately (i.e. independently). The non-convex sensing problems are transformed into convex optimization problems under certain practical conditions. Simulation results demonstrate the effectiveness of both the proposed frameworks and attest to their superior performances compared to contemporary strategies.