  Let $A$ be a sequence of positive integers and $P(A)$ be the set of all
integers which are the finite sum of distinct terms of $A$. For given positive
integers $u\in\{4,7,8\}\cup\{u:u\ge11\}$ and $v\ge 3u+5$ we know that $u+v+1$
is the critical value of $b_3$ such that there exists a sequence $A$ of
positive integers for which $P(A)=\mathbb{N}\backslash \{u<v<b_3<\cdots\}$. In
this paper, we obtain the critical value of $b_k$.

Number Theory