What if a specific neutrinoless double beta decay is absent?

We consider the seesaw model with two right-handed neutrinos, N-1 and N-2, the masses of which are hierarchical, and investigate their contribution to the neutrinoless double beta (0 nu beta beta) decay. Although the lepton number is broken by the Majorana masses of right-handed neutrinos, such decay processes can be absent in some cases. We present a possibility where the lighter N-1 gives a destructive contribution to the decay of active neutrinos by choosing the specific mixing elements of N-1, while N-2 is sufficiently heavy not to contribute to the 0 nu beta beta decay. In this case the mixing elements of N-1 in the charged current interaction are determined by its mass and the Majorana phase of active neutrinos. We then study the impacts of such a possibility on the direct search for N-1. In addition, we discuss the consequence of the case when the 0 nu beta beta decay in one specific nucleus is absent.