  A set of integers is sum-free if it contains no solution to the equation
$x+y=z$. We study sum-free subsets of the set of integers $[n]=\{1,\ldots,n\}$
for which the integer $2n+1$ cannot be represented as a sum of their elements.
We prove a bound of $O(2^{n/3})$ on the number of these sets, which matches, up
to a multiplicative constant, the lower bound obtained by considering all
subsets of $B_n = \{ \lceil \frac{2}{3}(n+1) \rceil, \ldots, n \}$. A main
ingredient in the proof is a stability theorem saying that if a subset of $[n]$
of size close to $|B_n|$ contains only a few subsets that contradict the
sum-freeness or the forbidden sum, then it is almost contained in $B_n$. Our
results are motivated by the question of counting symmetric complete sum-free
subsets of cyclic groups of prime order. The proofs involve Freiman's $3k-4$
theorem, Green's arithmetic removal lemma, and structural results on
independent sets in hypergraphs.
