Numerical semigroups are the subsemigroups of the set of natural numbers that are cofinite and contain $0$. Let $S$ be a numerical semigroup and $c$ be the smallest number such that $S$ is the union of a finite subset of $[0,c]$ and the integer interval $[c,\infty)$. Wilf's conjecture states that the density of elements of $S$ in the interval $[0,c]$ is at least equal to $1/d$, where $d$ is the dimension of the numerical semigroup $S$. We show the partial result that if $m$ is the smallest positive element of a numerical semigroup $S$ and if $S$ contains at least $\sqrt{3m}$ elements in $[m,2m)$ then it satisfies Wilf's conjecture. We discuss some implications of this, including the result that Wilf's conjecture holds for almost every numerical semigroup in a certain asymptotic sense. This is a joint work with Manuel Delgado and Claude Marion. 
 

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