Gaps in semigroups

In this paper we investigate the behaviour of the gaps   in numerical semigroups. We will give an explicit formula for the iith gap of a semigroup generated by k+1k+1 consecutive integers (generalizing a result due to Brauer) as well as for a special numerical semigroup of three generators. It is also proved that the number of gaps of the numerical semigroup generated by integers p and q   with g.c.d.(p,q)=1g.c.d.(p,q)=1, in the interval [pq-(k+1)(p+q),…,pq-k(p+q)][pq-(k+1)(p+q),…,pq-k(p+q)] is equals to2(k+1)+kqp+kpqfor eachk=1,…,pqp+q-1.We actually give two proofs of the latter result, the first one uses the so-called Apery sets and the second one is an application of the well-known Pick's theorem.