The thermoelectric properties of unintentionally n-doped core GaN/AlGaN core/shell N-face nanowires are reported. We found that the temperature dependence of the electrical conductivity is consistent with thermally activated carriers with two distinctive donor energies. The Seebeck coefficient of GaN/AlGaN nanowires is more than twice as large as that for the GaN nanowires alone. However, an outer layer of GaN deposited onto the GaN/AlGaN core/shell nanowires decreases the Seebeck coefficient at room temperature, while the temperature dependence of the electrical conductivity remains the same. We attribute these observations to the formation of an electron gas channel within the heavily-doped GaN core of the GaN/AlGaN nanowires. The room-temperature thermoelectric power factor for the GaN/AlGaN nanowires can be four times higher than the GaN nanowires. Selective doping in bandgap engineered core/shell nanowires is proposed for enhancing the thermoelectric power.