A385592 Values of u in the quartets (2,u,v,w); i.e., values of u for solutions to 2*(2+u) = v*(v+w), in positive integers, with and v>m, sorted by nondecreasing values of u; see Comments.

4, 7, 8, 10, 10, 12, 13, 13, 14, 16, 16, 18, 18, 19, 19, 20, 22, 22, 22, 23, 24, 25, 25, 26, 26, 28, 28, 28, 28, 30, 31, 31, 32, 33, 33, 34, 34, 34, 34, 36, 37, 37, 38, 38, 38, 40, 40, 40, 40, 42, 42, 43, 43, 43, 43, 44, 46, 46, 46, 46, 47, 48, 48, 49, 49

Offset: 1

Author

Clark Kimberling, Jul 04 2025

Keywords

• nonn

Comments

A 4-tuple (m,u,v,w) is a quartet if m,u,v,w are positive integers such that mA385182.

Examples

			First 30 quartets (2,u,v,w):
  m    u   v   w
  2    4   3   1
  2    7   3   3
  2    8   4   1
  2   10   3   5
  2   10   4   2
  2   12   4   3
  2   13   3   7
  2   13   5   1
  2   14   4   4
  2   16   3   9
  2   16   4   5
  2   18   4   6
  2   18   5   3
  2   19   3  11
  2   19   6   1
  2   20   4   7
  2   22   3  13
  2   22   4   8
  2   22   6   2
  2   23   5   5
  2   24   4   9
  2   25   3  15
  2   25   6   3
  2   26   4  10
  2   26   7   1
  2   28   3  17
  2   28   4  11
  2   28   5   7
  2   28   6   4
  2   30   4  12
2(2+16) = 3(3+9) = 4(4+5), so (2,16,3,9) and (2,16,4,5) are rows.
		

Crossrefs

Cf. A385182, A385592, A385593, A385594.

Programs

Mathematica

Clear[solnsM];
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], {#[[1]], #[[2]]} =!= {#[[3]], #[[4]]} &]]]], #[[1]] == m &]];
TableForm[solns = solnsM[2, 140], TableHeadings -> {None, {"m", "u", "v", "w"}}]
aa = Flatten[solns]
Map[#[[2]] &, solns]    (* u, A385592 *)
Map[#[[3]] &, solns]    (* v, A385593 *)
Map[#[[4]] &, solns]    (* w, A385594 *)
(* Peter J. C. Moses, Jun 15 2025 *)