A385182 Values of u in the quartets (1,u,v,w); i.e., values of u for solutions to (1+u) = v*(v+w), in positive integers, with v>1, sorted by nondecreasing values of u; see Comments.

5, 7, 9, 11, 11, 13, 14, 15, 17, 17, 19, 19, 20, 21, 23, 23, 23, 25, 26, 27, 27, 29, 29, 29, 31, 31, 32, 33, 34, 35, 35, 35, 37, 38, 39, 39, 39, 41, 41, 41, 43, 43, 44, 44, 45, 47, 47, 47, 47, 49, 49, 50, 51, 51, 53, 53, 53, 54, 55, 55, 55, 56, 57, 59, 59

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=1.

If m is a prime, then (u,v,w) = (m+2,m+1,m-1) is the first solution (in the defined ordering of triples).

u >= 1 appears A056924(u+1)-1 times. - Pontus von Brömssen, Jul 06 2025

Links

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

Example

First 30 quartets (1,u,v,w):
  m    u    v    w
  1    5    2    1
  1    7    2    2
  1    9    2    3
  1   11    2    4
  1   11    3    1
  1   13    2    5
  1   14    3    2
  1   15    2    6
  1   17    2    7
  1   17    3    3
  1   19    2    8
  1   19    4    1
  1   20    3    4
  1   21    2    9
  1   23    2   10
  1   23    3    5
  1   23    4    2
  1   25    2   11
  1   26    3    6
  1   27    2   12
  1   27    4    3
  1   29    2   13
  1   29    3    7
  1   29    5    1
  1   31    2   14
  1   31    4    4
  1   32    3    8
  1   33    2   15
  1   34    5    2
  1   35    2   16
1*(1+23) = 2*(2+10) = 3*(3+5) = 4*(4+2), so three of the rows are (1,23,2,10), (1,23,3,5), and (1,23,4,2).

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[1, 140], TableHeadings -> {None, {"m", "u", "v", "w"}}]
aa = Flatten[solns]
Map[#[[2]] &, solns]    (* u, A385182 *)
Map[#[[3]] &, solns]    (* v, A385183 *)
Map[#[[4]] &, solns]    (* w, A385184 *)
(*_Peter J.C.Moses_, Jun 15 2025*)

Crossrefs

Guide to related sequences:

m | u | v | w

--+---------+---------+--------

1 | A385182 | A385183 | A385184

2 | A385592 | A385593 | A385594

3 | A385595 | A385596 | A385597

4 | A385598 | A385599 | A385600

--+---------+---------+--------

Cf. A056924.

Sequence in context: A212191 A336122 A241853 * A385595 A165513 A002342

Adjacent sequences: A385179 A385180 A385181 * A385183 A385184 A385185

Keyword

nonn

Author

Clark Kimberling, Jun 23 2025

Status

approved