A semiprime is a number that factors into exactly two primes. The decision problem SEMIPRIME decides whether a given number is a semiprime.
It follows from the polynomiality of the AKS test that SEMIPRIME is both in NP and in co-NP: to verify "yes" write the input n as a n=pq and test that both p and q are primes, and to verify "no" test whether either n is a prime, or write n as a product of more than two nontrivial factors.
The prime example until 2002 of a decision problem that belongs to NP∩co-NP and was not known to be in P used to be PRIME, to decide whether a given number is a prime.
I wonder if more is known about the complexity of SEMIPRIME, in particular whether it is known to be in P? I have not been able to google any relevant information.