it follows that two algebroid branches are equivalent if and only if their Arf closure have the same value semigroup. Taking in account the properties of the value semigroup of an Arf ring, a numerical Arf semigroup is defined as a numerical semigroup $S \subseteq \mathbb{N}$ such that $S(s)-s$ is a semigroup, for each $s\in S$, where $S(s)=\left\{n \in S; n\geq s \right\}$. The multiplicity sequence of an Arf numerical semigroup $S=\left\{ s_0=0<s_1<s_2,\ldots \right\}$ is very easy to compute since it coincides with the sequence of the differences $s_{i+1}-s_i$. By focusing on the numerical properties that a sequence has to satisfy to be a multiplicity sequence, it is possible to study the Arf numerical semigroups with a combinatorial approach without referring to the ring theory context. In the first section of this paper we present a new procedure to compute all Arf numerical semigroups $S$ with a prescribed genus, that is the cardinality of $\mathbb{N} \setminus S$ (this problem was already addressed in [@algo]). Then we start dealing with the case $r>1$, considering the class of Arf good semigroups of $\mathbb{N}^r$. The definition of good semigroup was presented [@BDF] and it is based on the properties of the value semigroups 