Let S be a numerical semigroup with multiplicity m(S). Then, S is called a second-level numerical semigroup if x + y + z − m(S) ∈ S for every {x, y, z} ⊆ S \ {0}. In this paper, we present some algorithms to compute all the second-level numerical semigroups with multiplicity, genus, and a Frobenius fixed number. For m and r, which are positive integers, such that m < r and gcd(m, r) = 1, we show that there exists the minimal second-level numerical semigroup with multiplicity m containing r. We solve the Frobenius problem for these semigroups and show that they satisfy Wilf’s conjecture.
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