A386631 Values of u in the quartets (2, u, v, w) of type 3; i.e., values of u for solutions to 2*(2 - u) = v*(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

5, 6, 7, 8, 8, 8, 9, 10, 10, 11, 11, 11, 12, 12, 12, 13, 14, 14, 14, 14, 14, 15, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18, 18, 19, 20, 20, 20, 20, 20, 20, 21, 22, 22, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 25, 26, 26, 26, 26, 26, 26, 26, 27, 27, 27, 28

Offset

1,1

Comments

A 4-tuple (m, u, v, w) is a quartet of type 3 if m, u, v, w are distinct positive integers such that m < v and m*(m - u) = v*(v - w). Here, the values of u are arranged 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.

Links

Table of n, a(n) for n=1..66.

Example

First 20 quartets (2,u,v,w) of type 3:
   m    u    v    w
   2    5    6    7
   2    6    8    9
   2    7   10   11
   2    8    3    7
   2    8    4    7
   2    8   12   13
   2    9   14   15
   2   10    4    8
   2   10   16   17
   2   11    3    9
   2   11    6    9
   2   11   18   19
   2   12    4    9
   2   12    5    9
   2   12   20   21
   2   13   22   23
   2   14    3   11
   2   14    4   10
   2   14    6   10
   2   14    8   11
2(2-10) = 4(4-8), so (2, 10, 4, 8) is in the list.

Mathematica

ssolnsM[m_Integer?Positive, u_Integer?Positive] :=
  Module[{n = m  (m - u), nn, sgn, ds, tups}, If[n == 0, Return[{}]];
   sgn = Sign[n]; nn = Abs[n];
   ds = Divisors[nn];
   If[sgn > 0, ds = Select[ds, # < nn/# &]];
   tups = ({m, u, nn/#, nn/# - sgn  #} & /@ ds);
   Select[tups, #[[3]] > 1 && #[[4]] > 0 && #[[2]] =!= #[[4]] &&
   Length@DeleteDuplicates[#] == 4 &]];
(solns = Sort[Flatten[Map[solnsM[2, #] &, Range[2, 60]], 1]]) // ColumnForm
Map[#[[2]] &, solns] (*A386631*)
Map[#[[3]] &, solns] (*A387225*)
Map[#[[4]] &, solns] (*A387226*)
(* Peter J. C. Moses, Aug 22 2025 *)

Crossrefs

Cf. A385182 (type 1), A386218 (type 2), A385476 (type 3, m=1), A387225, A387226.

Sequence in context: A368126 A361472 A346172 * A114547 A217293 A201971

Adjacent sequences: A386628 A386629 A386630 * A386632 A386633 A386634

Keyword

nonn

Author

Clark Kimberling, Aug 22 2025

Status

approved