Almirón, PatricioMoyano Fernández, Julio-José2023-06-222023-06-222022-03-301607-360610.2989/16073606.2022.2041126https://hdl.handle.net/20.500.14352/71833We give an estimate of the minimal positive value of the Wilf function of a numerical semigroup in terms of its concentration. We describe necessary conditions for a numerical semigroup to have a negative Eliahou number in terms of its multiplicity, concentration and Wilf function. Also, we show new examples of numerical semigroups with a negative Eliahou number satisfying the Wilf conjecture. In addition, we introduce the notion of highly dense numerical semigroup; this yields a new family of numerical semigroups satisfying the Wilf conjecture. Moreover, we use the Wilf function of a numerical semigroup to prove that the Eliahou number of a highly dense numerical semigroup is positive under certain additional hypothesis. These results provide new evidences in favour of the Wilf conjecture.spajournal articlehttps://doi.org/10.2989/16073606.2022.2041126open access512512.53Numerical semigroupCoin change problemWilf conjectureEliahou numberConcentration.ÁlgebraGrupos (Matemáticas)1201 Álgebra