{"schema":"vela.problem-packet.v0.1","problem":121,"statement":"Let $F_{k}(N)$ be the size of the largest $A\\subseteq \\{1,\\ldots,N\\}$ such that the product of no $k$ many distinct elements of $A$ is a square. Is $F_5(N)=(1-o(1))N$? More generally, is $F_{2k+1}(N)=(1-o(1))N$?","status":"solved","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A013928","name":"Number of (positive) squarefree numbers < n.","terms":"0,1,2,3,3,4,5,6,6,6,7,8,8,9,10,11,11,12,12,13,13,14,15,16,16,16,17,17,17,18,19,20,20,21,22,23,23,24,25,26,26,27,28,29,29","url":"https://oeis.org/A013928"},{"id":"A028391","name":"a(n) = n - floor(sqrt(n)).","terms":"0,0,1,2,2,3,4,5,6,6,7,8,9,10,11,12,12,13,14,15,16,17,18,19,20,20,21,22,23,24,25,26,27,28,29,30,30,31,32,33,34,35,36,37,3","url":"https://oeis.org/A028391"},{"id":"A143301","name":"Decimal expansion of the Hall-Montgomery constant.","terms":"1,7,1,5,0,0,4,9,3,1,4,1,5,3,6,0,6,5,8,6,0,4,3,9,9,7,1,5,5,5,2,1,2,1,0,9,6,2,2,2,6,2,9,0,4,2,2,9,5,5,0,8,4,1,7,1,4,2,1,1,","url":"https://oeis.org/A143301"},{"id":"A360659","name":"a(n) is the minimum sum of a completely multiplicative sign sequence of length n.","terms":"0,1,0,-1,0,-1,0,-1,-2,-1,0,-1,-2,-3,-4,-3,-2,-3,-4,-5,-4,-5,-4,-5,-6,-5,-6,-7,-8,-9,-8,-9,-8,-7,-8,-7,-6,-7,-8,-7,-8,-9,","url":"https://oeis.org/A360659"},{"id":"A372306","name":"Cardinality of the largest subset of {1,...,n} such that no three distinct elements of this subset multiply to a square.","terms":"1,2,3,4,5,5,6,6,6,7,8,8,9,10,10,10,11,11,12,12,13,13,14,15,15,16,17,18,19,19,20,20,20,21,21,21,22,23,23,24,25,26,27,28,2","url":"https://oeis.org/A372306"},{"id":"A373114","name":"Cardinality of the largest subset of {1,...,n} such that no odd number of terms from this subset multiply to a square.","terms":"0,1,2,2,3,3,4,5,5,5,6,7,8,9,9,9,10,11,12,12,13,13,14,15,15,16,17,18,19,19,20,20,20,21,21,21,22,23,23,24,25,26,27,28,29,3","url":"https://oeis.org/A373114"},{"id":"A373178","name":"Cardinality of the largest subset of {1,...,n} such that no five distinct elements of this subset multiply to a square.","terms":"1,2,3,4,5,5,6,7,7,7,8,8,9,10,10,10,11,11,12,12,13,13,14,15,15,16,17,18,19,19,20,20,20,21,21,21,22,23,23,24,25,26,27,28,2","url":"https://oeis.org/A373178"},{"id":"A373319","name":"Denominator of the asymptotic density of numbers that are unitarily divided by n.","terms":"1,4,9,8,25,18,49,16,27,25,121,36,169,98,225,32,289,54,361,50,147,242,529,72,125,169,81,196,841,225,961,64,1089,289,1225,","url":"https://oeis.org/A373319"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}