A386631 -id:A386631 - cp's OEIS frontend

A387225 Values of v in the quartets (2, u, v, w) of type 3; i.e., values of v for solutions to 2(2 - u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see A386631.

6, 8, 10, 3, 4, 12, 14, 4, 16, 3, 6, 18, 4, 5, 20, 22, 3, 4, 6, 8, 24, 26, 4, 7, 28, 3, 5, 6, 10, 30, 4, 8, 32, 34, 3, 4, 6, 9, 12, 36, 38, 4, 5, 8, 10, 40, 3, 6, 7, 14, 42, 4, 11, 44, 46, 3, 4, 6, 8, 12, 16, 48, 5, 10, 50, 4, 13, 52, 3, 6, 9, 18, 54, 4, 7

Offset: 1

Clark Kimberling, Aug 22 2025

Cf. A386631.

A385476 Values of u in the quartets (1, u, v, w) of type 3; i.e., values of u for solutions to (1 - 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, 7, 7, 9, 9, 10, 11, 11, 13, 13, 13, 13, 15, 15, 16, 16, 17, 17, 17, 19, 19, 19, 19, 21, 21, 21, 21, 22, 22, 23, 23, 25, 25, 25, 25, 25, 25, 26, 27, 27, 28, 28, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 33, 33, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 37, 37

Offset: 1

Clark Kimberling, Aug 16 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 = 1.

			First 20 quartets (1,u,v,w) of type 3:
   m    u    v    w
   1    5    2    4
   1    7    2    5
   1    7    3    5
   1    9    2    6
   1    9    4    6
   1   10    3    6
   1   11    2    7
   1   11    5    7
   1   13    2    8
   1   13    3    7
   1   13    4    7
   1   13    6    8
   1   15    2    9
   1   15    7    9
   1   16    3    5
   1   16    3    8
   1   17    2   10
   1   17    4    8
   1   17    8   10
   1   19    2   11
1(1-11) = 5(5-7), so (1, 11, 5, 7) is in the list.

Cf. A385182 (type 1), A386218 (type 2), A386631, A385246.

solnsM[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[1, #] &, Range[2, 30]], 1]]) // ColumnForm
Map[#[[2]] &, solns] (*A385476*)
Map[#[[3]] &, solns] (*A163870*)
Map[#[[4]] &, solns] (*A385246*)
(* Peter J. C. Moses, Aug 22 2025 *)

cp's OEIS Frontend

A387225 Values of v in the quartets (2, u, v, w) of type 3; i.e., values of v for solutions to 2(2 - u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see A386631.

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

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