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

Author

Clark Kimberling, Aug 22 2025

Keywords

• nonn
• new

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.

Examples

			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.
		

Crossrefs

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

Programs

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 *)