A385884 Values of u in the quartets (2, u, v, w) of type 2; i.e., values of u for solutions to 2*(2 + u) = v*(v - w), in positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.
1, 3, 4, 5, 6, 7, 7, 8, 8, 9, 10, 10, 11, 12, 12, 13, 13, 13, 14, 14, 15, 16, 16, 16, 17, 18, 18, 18, 19, 19, 19, 20, 20, 21, 22, 22, 22, 23, 23, 24, 24, 25, 25, 25, 26, 26, 26, 27, 28, 28, 28, 28, 28, 29, 30, 30, 31, 31, 31, 32, 32, 33, 33, 33, 34, 34, 34
Offset: 1
Keywords
Examples
First 20 quartets (2,u,v,w) of type 2: m u v w 2 1 6 5 2 3 10 9 2 4 12 11 2 5 14 13 2 6 16 15 2 7 6 3 2 7 18 17 2 8 5 1 2 8 20 19 2 9 22 21 2 10 8 5 2 10 24 23 2 11 26 25 2 12 7 3 2 12 28 27 2 13 6 1 2 13 10 7 2 13 30 29 2 14 8 4 2 14 32 31 2 (2 +4) = 12 (12 - 11), so (2,4,12,11) is in the list.
Programs
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Mathematica
solnsM[m_, max_] := Module[{ans = {}, rhs = {}, u, v, w, lhs, matching}, Do[Do[AppendTo[rhs, {v*(v - w), v, w}], {w, max}], {v, m*(m + max)}]; rhs = GatherBy[rhs, First]; Do[lhs = m*(m + u); matching = Select[rhs, #[[1, 1]] == lhs &]; If[Length[matching] > 0, Do[AppendTo[ans, Map[{m, u, #[[2]], #[[3]]} &, matching[[1]]]], {i, Length[matching]}]], {u, max}]; ans = Flatten[ans, 1]; Select[ Union[Map[Sort[{#, RotateLeft[#, 2]}][[1]] &, Sort[Select[DeleteDuplicates[ans], Length[Union[#]] == 4 &]]]], #[[1]] == m &]]; TableForm[solns = solnsM[2, 100], TableHeadings -> {None, {"m", "u", "v", "w"}}] u1 = Map[#[[2]] &, solns] (*u, A385884 *) v1 = Map[#[[3]] &, solns] (*v, A386216 *) w1 = Map[#[[4]] &, solns] (*w, A386982 *) (* Peter J. C. Moses, Jun 15 2025 *)
Comments