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

1,1

Comments

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

Links

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

Example

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.

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*)

Crossrefs

Cf. A385182, A385592, A385599, A385600.

Sequence in context: A046344 A116366 A315849 * A138890 A189082 A171562

Adjacent sequences: A385595 A385596 A385597 * A385599 A385600 A385601

Keyword

nonn

Author

Clark Kimberling, Jul 10 2025

Status

approved