{"schema":"vela.problem-packet.v0.1","problem":242,"statement":"For every $n&#62;2$ there exist distinct integers $1\\leq x&#60;y&#60;z$ such that\\[\\frac{4}{n} = \\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}.\\]","status":"open","seam":"sealed","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[{"id":"att_2b154124e550b938","kind":"transfer","claim":"Transfer from #306: verified 4/n=1/x+1/y+1/z (1<=x<y<z) for every n=3..300, 0 failures (Opus re-checked 298 stored witnesses exactly).","grade":"verified_reduction","gateStatus":"needs_verification","superseded":false}],"velaLean":[],"oeis":[{"id":"A073101","name":"Number of integer solutions (x,y,z) to 4/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z.","terms":"0,0,1,1,2,5,5,6,4,9,7,15,4,14,33,22,4,21,9,30,25,22,19,45,10,17,25,36,7,72,17,62,27,22,59,69,9,29,67,84,7,77,12,56,87,39","url":"https://oeis.org/A073101"},{"id":"A075245","name":"x-value of the solution (x,y,z) to 4/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z and having the largest z-value. The y and z components are in A075246 and A075247.","terms":"1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,9,9,9,9,10,10,10,10,11,11,11,11,12,12,12,12,13,14,13,13,14,14,","url":"https://oeis.org/A075245"},{"id":"A075246","name":"y-value of the solution (x,y,z) to 4/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z and having the largest z-value. The x and z components are in A075245 and A075247.","terms":"4,3,4,7,15,7,10,16,34,13,18,29,61,21,30,46,96,31,43,67,139,43,60,92,190,57,78,121,249,73,100,154,316,91,124,191,391,111,","url":"https://oeis.org/A075246"},{"id":"A075247","name":"Largest possible z-value of an integer solution (x,y,z) to 4/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z. The x and y components are in A075245 and A075246.","terms":"12,6,20,42,210,42,90,240,1122,156,468,812,3660,420,510,2070,9120,930,1806,4422,19182,1806,2100,8372,35910,3192,9048,1452","url":"https://oeis.org/A075247"},{"id":"A075248","name":"Number of solutions (x,y,z) to 5/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z.","terms":"0,1,2,1,1,3,5,9,6,3,12,5,18,15,10,5,21,11,22,18,15,8,55,30,15,20,43,20,45,5,24,35,23,36,53,10,21,52,62,6,62,12,73,69,16,","url":"https://oeis.org/A075248"},{"id":"A287116","name":"Nonsquare integers that cannot be represented in the form 4M-d, where (a*b)|M and d|(a+b) for some positive integers a,b.","terms":"288,336,4545","url":"https://oeis.org/A287116"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}