{"schema":"vela.problem-packet.v0.1","problem":374,"statement":"For any $m\\in \\mathbb{N}$, let $F(m)$ be the minimal $k\\geq 2$ (if it exists) such that there are $a_1&#60;\\cdots &#60;a_k=m$ with $a_1!\\cdots a_k!$ a square. Let $D_k=\\{ m : F(m)=k\\}$. What is the order of growth of $\\lvert D_k\\cap\\{1,\\ldots,n\\}\\rvert$ for $3\\leq k\\leq 6$? For example, is it true that $\\lvert D_6\\cap \\{1,\\ldots,n\\}\\rvert \\gg n$?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A387184","name":"Numbers d such that a!*b!*c!*d! is a perfect square for some 1<=a<b<c<d.","terms":"6,8,9,10,12,14,15,16,18,20,21,22,24,25,26,27,28,30,32,33,34,35,36,38,39,40,42,44,45,48,49,50,51,52,54,55,56,57,60,62,63,","url":"https://oeis.org/A387184"},{"id":"A388851","name":"Numbers c such that a! * b! * c! is a perfect square for some 1 <= a < b < c.","terms":"4,6,8,9,10,16,18,20,24,25,28,30,32,35,36,45,49,50,54,63,64,70,72,77,80,81,96,98,100,112,120,121,125,126,128,140,144,150,","url":"https://oeis.org/A388851"},{"id":"A389117","name":"Decimal expansion of the sum of the distinct entries of 1/A055204(n)^(1/2).","terms":"3,7,0,9,7,5,1,2,3,3,8,9,9,2,2,2,3,6,8,8,8,7,9,6,1,4,6,6,8,4,2,0,7,8,8,8,2,1,7,4,4,3,6,4,5,0,6,4,1,6,9,7,6,6,8,0,9,7,8,9,","url":"https://oeis.org/A389117"},{"id":"A389148","name":"Composite numbers e>6 for which there is no 1<=a<b<c<d<e for which a!b!c!d!e! is a perfect square.","terms":"527,611,713,731,779,893,923,1003,1037,1271,1273,1343,1349,1357,1411,1469,1591,1643,1679,1781,1919,1927,1943,1957,2033,20","url":"https://oeis.org/A389148"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}