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

1,1

Comments

A 4-tuple (m,u,v,w) is a quartet if m,u,v,w are positive integers such that m<v 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=2; for m=1, see A385182.

Links

Table of n, a(n) for n=1..65.

Example

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.

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 *)

Crossrefs

Cf. A385182, A385592, A385593, A385594.

Sequence in context: A076680 A235623 A001074 * A214206 A026316 A283767

Adjacent sequences: A385589 A385590 A385591 * A385593 A385594 A385595

Keyword

nonn

Author

Clark Kimberling, Jul 04 2025

Status

approved