A385595 The u sequence in quartets (3,u,v,w); i.e., values of u for solutions to 3*(3+u) = v*(v+w), in positive integers, with u,v>=3 and u>=m, sorted by nondecreasing values of u; see Comments.

5, 7, 9, 11, 12, 13, 13, 15, 17, 17, 17, 18, 19, 21, 21, 21, 22, 23, 25, 25, 25, 27, 27, 27, 29, 29, 29, 30, 31, 32, 32, 33, 33, 33, 35, 36, 37, 37, 37, 37, 37, 39, 39, 39, 41, 41, 41, 42, 42, 43, 45, 45, 45, 45, 46, 47, 47, 47, 48, 49, 49, 49, 51, 51, 52

Offset: 1

Author

Clark Kimberling, Jul 07 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<=u, mA385182.

Examples

			First 30 quartets (3,u,v,w):
   m    u    v    w
   3    5    4    2
   3    7    5    1
   3    9    4    5
   3   11    6    1
   3   12    5    4
   3   13    4    8
   3   13    6    2
   3   15    6    3
   3   17    4   11
   3   17    5    7
   3   17    6    4
   3   18    7    2
   3   19    6    5
   3   21    4   14
   3   21    4   14
   3   21    6    6
   3   21    8    1
   3   22    5   10
   3   23    6    7
   3   25    4   17
   3   25    6    8
   3   25    7    5
   3   27    5   13
   3   27    6    9
   3   27    9    1
   3   29    4   20
   3   29    6   10
   3   29    8    4
   3   30    9    2
   3   31    6   11
3(3+13) = 4(4+8) = 6(6+2), so (3,13,4,8) and (3,13,6,2) are rows.
		

Crossrefs

Cf. A385182, A385592, A385596, A385597.

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[3, 140], TableHeadings -> {None, {"m", "u", "v", "w"}}]
aa = Flatten[solns]
Map[#[[2]] &, solns]    (* u, A385595 *)
Map[#[[3]] &, solns]    (* v, A385596 *)
Map[#[[4]] &, solns]    (* w, A385597 *)
(*Peter J.C.Moses, Jun 15 2025*)