Positioned numerical semigroups with maximal gender as function of multiplicity and Frobenius number

A $C$-semigroup (respectively a $D$-semigroup) is positioned numerical semigroup $S$ such that $\rm{g}(S)=\frac{\rm{F}(S)+\rm{m}(S)-1}{2}$ $\rm{g}(S)=\frac{\rm{F}(S)+\rm{m}(S)-2}{2}$). In this paper we study these semigroups giving formulas for the Frobenius number, pseudo-Frobenius and type. Furthermore, give algorithms computing whole set of $C$-semigroups $D$-semigroups.