Call a set $S\subseteq \{1,\dots ,n\}$ admissible if $(a,b)=1$ for all $a\ne b\in S$. Let$$G(n)=\underset{S\subseteq \{1,\dots ,n\}}{\max }\sum _{a\in S}a$$and$$H(n)=\sum _{p<n}p+n\pi (n^{1/2}).$$Is it true that$$G(n)>H(n)-n^{1+o(1)}?$$Is it true that, for every $k\ge 2$, if $n$ is sufficiently large then the admissible set which maximises $G(n)$ contains at least one integer with at least $k$ prime factors?