  Let $S\neq\mathbb N$ be a numerical semigroup generated by $e$ elements. In
his paper (A Circle-Of-Lights Algorithm for the "Money-Changing Problem", Amer.
Math. Monthly 85 (1978), 562--565), H.~S.~Wilf raised the following question:
Let $\Omega$ be the number of positive integers not contained in $S$ and $c-1$
the largest such element. Is it true that the fraction $\frac\Omega c$ of
omitted numbers is at most $1-\frac1e$?
  Let $B\subseteq\mathbb N^{e-1}$ be the complement of an artinian $\mathbb
N^{e-1}$-ideal. Following a concept of A.~Zhai (An asymptotic result concerning
a question of Wilf, arXiv:1111.2779v1 [math.CO]) we relate Wilf's problem to a
more general question about the weight distribution on $B$ with respect to a
positive weight vector. An affirmative answer is given in special cases,
similar to those considered by R.~Fr\"oberg, C.~Gottlieb, R.~H\"aggkvist (On
numerical semigroups, Semigroup Forum, Vol.~35, Issue 1, 1986/1987, 63--83) for
Wilf's question.
