A386631 Values of u in the quartets (2, u, v, w) of type 3; i.e., values of u for solutions to 2(2 - u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.
5, 6, 7, 8, 8, 8, 9, 10, 10, 11, 11, 11, 12, 12, 12, 13, 14, 14, 14, 14, 14, 15, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18, 18, 19, 20, 20, 20, 20, 20, 20, 21, 22, 22, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 25, 26, 26, 26, 26, 26, 26, 26, 27, 27, 27, 28
Offset: 1
Examples
First 20 quartets (2,u,v,w) of type 3: m u v w 2 5 6 7 2 6 8 9 2 7 10 11 2 8 3 7 2 8 4 7 2 8 12 13 2 9 14 15 2 10 4 8 2 10 16 17 2 11 3 9 2 11 6 9 2 11 18 19 2 12 4 9 2 12 5 9 2 12 20 21 2 13 22 23 2 14 3 11 2 14 4 10 2 14 6 10 2 14 8 11 2(2-10) = 4(4-8), so (2, 10, 4, 8) is in the list.
Programs
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Mathematica
ssolnsM[m_Integer?Positive, u_Integer?Positive] := Module[{n = m (m - u), nn, sgn, ds, tups}, If[n == 0, Return[{}]]; sgn = Sign[n]; nn = Abs[n]; ds = Divisors[nn]; If[sgn > 0, ds = Select[ds, # < nn/# &]]; tups = ({m, u, nn/#, nn/# - sgn #} & /@ ds); Select[tups, #[[3]] > 1 && #[[4]] > 0 && #[[2]] =!= #[[4]] && Length@DeleteDuplicates[#] == 4 &]]; (solns = Sort[Flatten[Map[solnsM[2, #] &, Range[2, 60]], 1]]) // ColumnForm Map[#[[2]] &, solns] (*A386631*) Map[#[[3]] &, solns] (*A387225*) Map[#[[4]] &, solns] (*A387226*) (* Peter J. C. Moses, Aug 22 2025 *)
Comments