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

1,1

Comments

A 4-tuple (m,u,v,w) is a quartet if m,u,v,w are positive integers such that m<=u, 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=3; for m=1, see A385182.

Links

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

Example

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.

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*)

Crossrefs

Cf. A385182, A385592, A385596, A385597.

Sequence in context: A336122 A241853 A385182 * A165513 A002342 A080353

Adjacent sequences: A385592 A385593 A385594 * A385596 A385597 A385598

Keyword

nonn

Author

Clark Kimberling, Jul 07 2025

Status

approved