Problem 879

Erdős Problem #879: declared status 'open'. Formalized: no. Call a set $S\subseteq \{1,\ldots,n\}$ admissible if $(a,b)=1$ for all $a\neq b\in S$. Let$$G(n) = \max_{S\subseteq \{1,\ldots,n\}} \sum_{a\in S}a$$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 $k\geq 2$, if $n$ is sufficiently large then the admissible set which maximises $G(n)$ contains at least one integer with at least $k$ prime factors? Prize: no. OEIS: A186736. Tags: number theory.