A385476 -id:A385476 - cp's OEIS frontend

A385246 Values of w in the quartets (1, u, v, w) of type 3; i.e., values of w for solutions to 1(1 - u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see A385476.

4, 5, 5, 6, 6, 6, 7, 7, 8, 7, 7, 8, 9, 9, 8, 8, 10, 8, 10, 11, 9, 9, 11, 12, 9, 9, 12, 10, 10, 13, 13, 14, 11, 10, 10, 11, 14, 10, 15, 15, 12, 12, 16, 11, 11, 16, 17, 13, 11, 11, 13, 17, 18, 12, 12, 18, 14, 14, 19, 19, 12, 12, 20, 15, 13, 12, 13, 15, 20, 21

Offset: 1

Clark Kimberling, Aug 22 2025

Cf. A385476.

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.

Original entry on oeis.org

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

Clark Kimberling, Aug 22 2025

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.

			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.

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

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 *)

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A385246 Values of w in the quartets (1, u, v, w) of type 3; i.e., values of w for solutions to 1(1 - u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see A385476.

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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.

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