  Let $p_1=2, p_2=3, p_3=5, \ldots$ be the consecutive prime numbers, $S_n$ the
numerical semigroup generated by the primes not less than $p_n$ and $u_n$ the
largest irredundant generator of $S_n$. We will show, that
  $\bullet$ $u_n\sim3p_n$.
  Similarly, for the largest integer $f_n$ not contained in $S_n$, by
computational evidence we suspect that
  $\bullet$ $f_n$ is an odd number for $n\geq5$ and
  $\bullet$ $f_n\sim3p_n$; further
  $\bullet$ $4p_n>f_{n+1}$ for $n\geq1$.
  If $f_n$ is odd for large $n$, then $f_n\sim3p_n$. In case $f_n\sim3p_n$
every large even integer $x$ is the sum of two primes. If $4p_n>f_{n+1}$ for
$n\geq1$, then the Goldbach conjecture holds true.
  Further, Wilf's question in [12] has a positive answer for the semigroups
$S_n$.
