{"schema":"vela.problem-packet.v0.1","problem":168,"statement":"Let $F(N)$ be the size of the largest subset of $\\{1,\\ldots,N\\}$ which does not contain any set of the form $\\{n,2n,3n\\}$. What is\\[ \\lim_{N\\to \\infty}\\frac{F(N)}{N}?\\]Is this limit irrational?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A004059","name":"a(n) gives position of first n in A057561.","terms":"1,2,4,5,6,8,9,11,13,14,15,17,18,20,22,23,24,26,28,29,30,32,34,35,36,38,40,41,42,43,45,47,48,50,51,53,55,56,57,59,60,61,6","url":"https://oeis.org/A004059"},{"id":"A057561","name":"Size of the largest set encompassing no {x, 2x, 3x} contained in D(n) = the first n 3-smooth numbers {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, ...} (A003586).","terms":"1,2,2,3,4,5,5,6,7,7,8,8,9,10,11,11,12,13,13,14,14,15,16,17,17,18,18,19,20,21,21,22,22,23,24,25,25,26,26,27,28,29,30,30,3","url":"https://oeis.org/A057561"},{"id":"A094708","name":"Size of the smallest set hitting all {x, 2x, 3x} contained in D(n) = the first n 3-smooth numbers {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27,...} (A003586).","terms":"0,0,1,1,1,1,2,2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,7,7,8,8,9,9,9,9,10,10,11,11,11,11,12,12,13,13,13,13,13,14,14,15,15,15,16,16,","url":"https://oeis.org/A094708"},{"id":"A386439","name":"Decimal expansion of the maximal density of a set of positive integers free of subsets of the form {n, 2n, 3n}.","terms":"8,0,0,9,6,5,7,5,5,0,0,6,5,5,8,9,8,9,0,9,0,4,2,0,3,2,6,3,8,8,0,8,2,4,1","url":"https://oeis.org/A386439"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}