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In this article, we study the quotients of numerical semigroups, generated bytwo coprime positive numbers, named (a,b) over d. We give formulae for theusual invariants of these semigroups, expressed in terms of continued fractionexpansions and Ostrowski-like numeration of some rationals, simply related toentries a, b, d. So, we obtain quadratic complexity algorithms to compute theseinvariants. As a consequence, we will prove that, for these numericalsemigroups, the type is always lower than the embedding dimension and deduceWilf's property. We also consider the '' reverse problem'' : given J, a finiteset of integers, we obtain an expression of all possible triplets (a, b, d),such that J is the set of minimal generators of (a,b) over d.

Quotients of numerical semigroups generated by two numbers