A385598 The u sequence in quartets (4,u,v,w); i.e., values of u for solutions to 4(4+u) = v(v+w), in positive integers, v>m, sorted by nondecreasing values of u; see Comments.

6, 8, 10, 11, 11, 14, 14, 16, 16, 17, 17, 18, 20, 20, 21, 22, 23, 23, 24, 24, 26, 26, 26, 26, 28, 29, 29, 30, 31, 31, 31, 32, 32, 32, 34, 35, 35, 36, 36, 36, 38, 38, 38, 38, 40, 40, 41, 41, 41, 41, 41, 42, 44, 44, 44, 45, 46, 46, 46, 47, 47, 48, 48, 50, 50

Offset: 1

Author

Clark Kimberling, Jul 10 2025

Keywords

• nonn

Comments

A 4-tuple (m,u,v,w) is a quartet if m,u,v,w are positive integers such that m>v and and m*(m+u) = v*(v+w), with the values of u in nondecreasing order. When there is more than one solution for given m and u, the values of v are arranged in increasing order. Here, m=4; for m=1, see A385182.

Examples

			First 30 quartets (4,u,v,w):
   m    u    v    w
   4    6    5    3
   4    8    6    2
   4   10    7    1
   4   11    5    7
   4   11    6    4
   4   14    6    6
   4   14    8    1
   4   16    5   11
   4   16    8    2
   4   17    6    8
   4   17    7    5
   4   18    8    3
   4   20    6   10
   4   20    8    4
   4   21    5   15
   4   22    8    5
   4   23    6   12
   4   23    9    3
   4   24    7    9
   4   24    8    6
   4   26    5   19
   4   26    6   14
   4   26    8    7
   4   26   19    2
   4   28    8    8
   4   29    6   16
   4   29   11    1
   4   30    8    9
   4   31    5   23
   4   31    7   13
4(4+16) = 5(5+11) = 8(8+2), so (4,16,5,11) and (4,16,8,2) are rows.
		

Crossrefs

Cf. A385182, A385592, A385599, A385600.

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[4, 140], TableHeadings -> {None, {"m", "u", "v", "w"}}]
aa = Flatten[solns]
Map[#[[2]] &, solns]    (* u, A385598 *)
Map[#[[3]] &, solns]    (* v, A385599 *)
Map[#[[4]] &, solns]    (* w, A385600 *)
(*Peter J.C.Moses, Jun 15 2025*)