  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.
