On a question of Eliahou and a conjecture of Wilf

To a numerical semigroup $S$, Eliahou associated a number $E(S)$ and proved that numerical semigroups for which the associated number is non negative satisfy Wilf's conjecture. The search for counterexamples for the conjecture of Wilf is therefore reduced to semigroups which have an associated negative Eliahou number. Eliahou mentioned $5$ numerical semigroups whose Eliahou number is $-1$. The examples were discovered by Fromentin who observed that these are the only ones with negative Eliahou number among the over $10^{13}$ numerical semigroups of genus up to $60$. We prove here that for any integer $n$ there are infinitely many numerical semigroups S such that $E(S)=n$, by explicitly giving families of such semigroups. We prove that all the semigroups in these families satisfy Wilf's conjecture, thus providing not previously known examples of semigroups for which the conjecture holds.