  Let S $\subseteq$ N be a numerical semigroup with multiplicity m = min(S \
{0}), conductor c = max(N \ S) + 1 and minimally generated by e elements. Let L
be the set of elements of S which are smaller than c. Wilf conjectured in 1978
that |L| is bounded below by c/e. We show here that if c $\le$ 3m, then S
satisfies Wilf's conjecture. Combined with a recent result of Zhai, this
implies that the conjecture is asymptotically true as the genus g(S) = |N \ S|
goes to infinity. One main tool in this paper is a classical theorem of
Macaulay on the growth of Hilbert functions of standard graded algebras.
