A385182 -id:A385182 - cp's OEIS frontend

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.

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 *)

A385593 The v sequence in quartets (2,u,v,w); see A385592.

Original entry on oeis.org

3, 3, 4, 3, 4, 4, 3, 5, 4, 3, 4, 4, 5, 3, 6, 4, 3, 4, 6, 5, 4, 3, 6, 4, 7, 3, 4, 5, 6, 4, 3, 6, 4, 5, 7, 3, 4, 6, 8, 4, 3, 6, 4, 5, 8, 3, 4, 6, 7, 4, 8, 3, 5, 6, 9, 4, 3, 4, 6, 8, 7, 4, 5, 3, 6, 4, 8, 3, 4, 6, 9, 5, 10, 4, 7, 8, 3, 6, 4, 3, 4, 5, 6, 8, 10, 4

Offset: 1

Clark Kimberling, Jul 04 2025

nonn

Cf. A385182, A385592, A385594.

A385594 The w sequence in quartets (2,u,v,w); see A385592.

Original entry on oeis.org

1, 3, 1, 5, 2, 3, 7, 1, 4, 9, 5, 6, 3, 11, 1, 7, 13, 8, 2, 5, 9, 15, 3, 10, 1, 17, 11, 7, 4, 12, 19, 5, 13, 9, 3, 21, 14, 6, 1, 15, 23, 7, 16, 11, 2, 25, 17, 8, 5, 18, 3, 27, 13, 9, 1, 19, 29, 20, 10, 4, 7, 21, 15, 31, 11, 22, 5, 33, 23, 12, 3, 17, 1, 24, 9

Offset: 1

Clark Kimberling, Jul 07 2025

nonn

Cf. A385182, A385592, A385593.

A385884 Values of u in the quartets (2, u, v, w) of type 2; i.e., values of u for solutions to 2(2 + u) = v(v - w), in positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 7, 8, 8, 9, 10, 10, 11, 12, 12, 13, 13, 13, 14, 14, 15, 16, 16, 16, 17, 18, 18, 18, 19, 19, 19, 20, 20, 21, 22, 22, 22, 23, 23, 24, 24, 25, 25, 25, 26, 26, 26, 27, 28, 28, 28, 28, 28, 29, 30, 30, 31, 31, 31, 32, 32, 33, 33, 33, 34, 34, 34

Offset: 1

Clark Kimberling, Aug 07 2025

nonn

A 4-tuple (m, u, v, w) is a quartet of type 2 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 2:
  m   u   v   w
  2   1   6   5
  2   3  10   9
  2   4  12  11
  2   5  14  13
  2   6  16  15
  2   7   6   3
  2   7  18  17
  2   8   5   1
  2   8  20  19
  2   9  22  21
  2  10   8   5
  2  10  24  23
  2  11  26  25
  2  12   7   3
  2  12  28  27
  2  13   6   1
  2  13  10   7
  2  13  30  29
  2  14   8   4
  2  14  32  31
2 (2 +4) = 12 (12 - 11), so (2,4,12,11) is in the list.

Cf. A385182 (type 1, m=1), A386630 (type 3, m=1).

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],
        Length[Union[#]] == 4 &]]]], #[[1]] == m &]];
TableForm[solns = solnsM[2, 100],
 TableHeadings -> {None, {"m", "u", "v", "w"}}]
u1 = Map[#[[2]] &, solns]  (*u, A385884 *)
v1 = Map[#[[3]] &, solns]  (*v, A386216 *)
w1 = Map[#[[4]] &, solns]  (*w, A386982 *)
(* Peter J. C. Moses, Jun 15 2025 *)

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 *)

A385596 The v sequence in quartets (3,u,v,w); see A385595.

Original entry on oeis.org

4, 5, 4, 6, 5, 4, 6, 6, 4, 5, 6, 7, 6, 4, 6, 8, 5, 6, 4, 6, 7, 5, 6, 9, 4, 6, 8, 9, 6, 5, 7, 4, 6, 9, 6, 9, 4, 5, 6, 8, 10, 6, 7, 9, 4, 6, 11, 5, 9, 6, 4, 6, 8, 9, 7, 5, 6, 10, 9, 4, 6, 12, 6, 9, 5, 11, 4, 6, 7, 8, 12, 9, 6, 4, 5, 6, 9, 10, 12, 6, 7, 9, 4, 6

Offset: 1

Clark Kimberling, Jul 15 2025

nonn

Cf. A385182, A385595.

A385597 The w sequence in quartets (3,u,v,w); see A385595.

Original entry on oeis.org

2, 1, 5, 1, 4, 8, 2, 3, 11, 7, 4, 2, 5, 14, 6, 1, 10, 7, 17, 8, 5, 13, 9, 1, 20, 10, 4, 2, 11, 16, 8, 23, 12, 3, 13, 4, 26, 19, 14, 7, 2, 15, 11, 5, 29, 16, 1, 22, 6, 17, 32, 18, 10, 7, 14, 25, 19, 5, 8, 35, 20, 1, 21, 9, 28, 4, 38, 22, 17, 13, 2, 10, 23, 41

Offset: 1

Clark Kimberling, Jul 10 2025

nonn

Cf. A385182, A385595.

cp's OEIS Frontend

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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A385593 The v sequence in quartets (2,u,v,w); see A385592.

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A385594 The w sequence in quartets (2,u,v,w); see A385592.

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A385884 Values of u in the quartets (2, u, v, w) of type 2; i.e., values of u for solutions to 2*(2 + u) = v*(v - w), in positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

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Examples

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Programs

Mathematica

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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Comments

Examples

Crossrefs

Programs

Mathematica

A385596 The v sequence in quartets (3,u,v,w); see A385595.

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Author

Keywords

Crossrefs

A385597 The w sequence in quartets (3,u,v,w); see A385595.

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Author

Keywords

Crossrefs

A385884 Values of u in the quartets (2, u, v, w) of type 2; i.e., values of u for solutions to 2(2 + u) = v(v - w), in positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.