In this study, we characterize all families of saturated numerical semigroups with multiplicity four. We also present some results about invariants of these semigroups.

In this paper, we will give some results about frobenius number, gaps, and determine number of arf closure of telescopic numerical semigroup k S such that 8,8 2, k S k x   where 1, k k   , 0,2, 4,...,2( 1) j k   and 8 2 (2 1) x k j     is odd integer number.