{"schema":"vela.problem-packet.v0.1","problem":879,"statement":"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?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A186736","name":"Maximum sum of relatively prime integers no larger than n.","terms":"1,3,6,8,13,13,20,24,30,30,41,41,54,54,55,63,80,80,99,99,103,103,126,126,146,146,159,164,193,193,224,235,235,235,238,238,","url":"https://oeis.org/A186736"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}