In this article in Section~2 we give an explicit description to compute the type sequence $mathrm{t}_1,ldots,mathrm{t}_{n}$ of a semigroup $Gamma$ generated by an arithmetic sequence (see 2.7); we show that the $i$-th term $mathrm{t}_i$ is equal to $1$ or to the type $au_Gamma$, depending on its position. In Section 3, for analytically irreducible ring $R$ with the branch sequence$R=R_0 subsetneq R_1 subsetneq ldotssubsetneq R_{m-1}subsetneq R_{m} =overline{R}$, starting from a result proved in [4] we give a characterization (see 3.6) of the ``Arf'' property using the type sequence of $R$ and of the rings $R_j$, $1leq jleq m-1$. Further, we prove (see 3.9, 3.10) some relations among the integers $ell^*(R)$ and $ell^*(R_j)$, $1leq jleq m-1$. These relations and a result of [6] allow us to obtain a new characterization (see 3.12) of semigroup rings of minimal multiplicity with $ell^*(R)leq au(R)$ in terms of the Arf property, type sequences and relations between $ell^*(R)$ and $ell^*(R_j)$, $1leq jleq m-1$.
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