A385182 -id:A385182 - cp's OEIS frontend

A385599 The v sequence in quartets (4,u,v,w); see A385182.

5, 6, 7, 5, 6, 6, 8, 5, 8, 6, 7, 8, 6, 8, 5, 8, 6, 9, 7, 8, 5, 6, 8, 10, 8, 6, 11, 8, 5, 7, 10, 6, 8, 9, 8, 6, 12, 5, 8, 10, 6, 7, 8, 12, 8, 11, 5, 6, 9, 10, 12, 8, 6, 8, 12, 7, 5, 8, 10, 6, 12, 8, 13, 6, 8, 9, 12, 5, 10, 11, 7, 8, 14, 6, 12, 8, 5, 6, 8, 10

Offset: 1

Clark Kimberling, Jul 12 2025

nonn

Cf. A385182, A385598.

A385600 The w sequence in quartets (4,u,v,w); see A385182.

Original entry on oeis.org

3, 2, 1, 7, 4, 6, 1, 11, 2, 8, 5, 3, 10, 4, 15, 5, 12, 3, 9, 6, 19, 14, 7, 2, 8, 16, 1, 9, 23, 13, 4, 18, 10, 7, 11, 20, 1, 27, 12, 6, 22, 17, 13, 2, 14, 5, 31, 24, 11, 8, 3, 15, 26, 16, 4, 21, 35, 17, 10, 28, 5, 18, 3, 30, 19, 15, 6, 39, 12, 9, 25, 20, 2

Offset: 1

Clark Kimberling, Jul 12 2025

nonn

Cf. A385182, A385598.

A385183 The v sequence in quartets (1,u,v,w); see A385182.

Original entry on oeis.org

2, 2, 2, 2, 3, 2, 3, 2, 2, 3, 2, 4, 3, 2, 2, 3, 4, 2, 3, 2, 4, 2, 3, 5, 2, 4, 3, 2, 5, 2, 3, 4, 2, 3, 2, 4, 5, 2, 3, 6, 2, 4, 3, 5, 2, 2, 3, 4, 6, 2, 5, 3, 2, 4, 2, 3, 6, 5, 2, 4, 7, 3, 2, 2, 3, 4, 5, 6, 2, 3, 7, 2, 4, 5, 2, 3, 6, 2, 4, 3, 2, 5, 7, 2, 3, 4

Offset: 1

Clark Kimberling, Jun 23 2025

nonn

Cf. A385182.

A385184 The w sequence in quartets (1,u,v,w); see A385182.

Original entry on oeis.org

1, 2, 3, 4, 1, 5, 2, 6, 7, 3, 8, 1, 4, 9, 10, 5, 2, 11, 6, 12, 3, 13, 7, 1, 14, 4, 8, 15, 2, 16, 9, 5, 17, 10, 18, 6, 3, 19, 11, 1, 20, 7, 12, 4, 21, 22, 13, 8, 2, 23, 5, 14, 24, 9, 25, 15, 3, 6, 26, 10, 1, 16, 27, 28, 17, 11, 7, 4, 29, 18, 2, 30, 12, 8, 31

Offset: 1

Clark Kimberling, Jun 26 2025

nonn

Cf. A385182.

A385592 Values of u in the quartets (2,u,v,w); i.e., values of u for solutions to 2(2+u) = v(v+w), in positive integers, with and v>m, sorted by nondecreasing values of u; see Comments.

Original entry on oeis.org

4, 7, 8, 10, 10, 12, 13, 13, 14, 16, 16, 18, 18, 19, 19, 20, 22, 22, 22, 23, 24, 25, 25, 26, 26, 28, 28, 28, 28, 30, 31, 31, 32, 33, 33, 34, 34, 34, 34, 36, 37, 37, 38, 38, 38, 40, 40, 40, 40, 42, 42, 43, 43, 43, 43, 44, 46, 46, 46, 46, 47, 48, 48, 49, 49

Offset: 1

Clark Kimberling, Jul 04 2025

nonn

A 4-tuple (m,u,v,w) is a quartet if m,u,v,w are positive integers such that mA385182.

			First 30 quartets (2,u,v,w):
  m    u   v   w
  2    4   3   1
  2    7   3   3
  2    8   4   1
  2   10   3   5
  2   10   4   2
  2   12   4   3
  2   13   3   7
  2   13   5   1
  2   14   4   4
  2   16   3   9
  2   16   4   5
  2   18   4   6
  2   18   5   3
  2   19   3  11
  2   19   6   1
  2   20   4   7
  2   22   3  13
  2   22   4   8
  2   22   6   2
  2   23   5   5
  2   24   4   9
  2   25   3  15
  2   25   6   3
  2   26   4  10
  2   26   7   1
  2   28   3  17
  2   28   4  11
  2   28   5   7
  2   28   6   4
  2   30   4  12
2(2+16) = 3(3+9) = 4(4+5), so (2,16,3,9) and (2,16,4,5) are rows.

Cf. A385182, A385592, A385593, A385594.

Clear[solnsM];
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], {#[[1]], #[[2]]} =!= {#[[3]], #[[4]]} &]]]], #[[1]] == m &]];
TableForm[solns = solnsM[2, 140], TableHeadings -> {None, {"m", "u", "v", "w"}}]
aa = Flatten[solns]
Map[#[[2]] &, solns]    (* u, A385592 *)
Map[#[[3]] &, solns]    (* v, A385593 *)
Map[#[[4]] &, solns]    (* w, A385594 *)
(* Peter J. C. Moses, Jun 15 2025 *)

A386218 Values of u in the quartets (1, u, v, w) of type 2; i.e., values of u for solutions to (1 + u) = v(v - w), in positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

Original entry on oeis.org

7, 9, 11, 13, 14, 15, 17, 17, 19, 20, 21, 23, 23, 23, 25, 26, 27, 27, 29, 29, 31, 31, 32, 33, 34, 35, 35, 35, 37, 38, 39, 39, 39, 41, 41, 43, 43, 44, 44, 45, 47, 47, 47, 47, 49, 49, 50, 51, 51, 53, 53, 53, 54, 55, 55, 56, 57, 59, 59, 59, 59, 59, 61, 62, 62

Offset: 1

Clark Kimberling, Jul 29 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 = 1.

			First 20 quartets (1,u,v,w) of type 2:
  m   u   v   w
  1   7   4   2
  1   9   5   3
  1  11   6   4
  1  13   7   5
  1  14   5   2
  1  15   8   6
  1  17   6   3
  1  17   9   7
  1  19  10   8
  1  20   7   4
  1  21  11   9
  1  23   6   2
  1  23   8   5
  1  23  12  10
  1  25  13  11
  1  26   9   6
  1  27   7   3
  1  27  14  12
  1  29  10   7
  1  29  15  13
Example : 1 (1 + 17) = 6 (6 - 3), so (1, 17, 6, 3) is in the list.

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

Clear[solnsM];
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[1, 100],
TableHeadings -> {None, {"m", "u", "v", "w"}}]
u1 = Map[#[[2]] &, solns]  (*u, A386218 *)
v1 = Map[#[[3]] &, solns]  (*v, A385883 *)
w1 = Map[#[[4]] &, solns]  (*w, A386630 *)
(* Peter J. C. Moses, Jun 15 2025  *)

A385595 The u sequence in quartets (3,u,v,w); i.e., values of u for solutions to 3(3+u) = v(v+w), in positive integers, with u,v>=3 and u>=m, sorted by nondecreasing values of u; see Comments.

Original entry on oeis.org

5, 7, 9, 11, 12, 13, 13, 15, 17, 17, 17, 18, 19, 21, 21, 21, 22, 23, 25, 25, 25, 27, 27, 27, 29, 29, 29, 30, 31, 32, 32, 33, 33, 33, 35, 36, 37, 37, 37, 37, 37, 39, 39, 39, 41, 41, 41, 42, 42, 43, 45, 45, 45, 45, 46, 47, 47, 47, 48, 49, 49, 49, 51, 51, 52

Offset: 1

Clark Kimberling, Jul 07 2025

nonn

A 4-tuple (m,u,v,w) is a quartet if m,u,v,w are positive integers such that m<=u, mA385182.

			First 30 quartets (3,u,v,w):
   m    u    v    w
   3    5    4    2
   3    7    5    1
   3    9    4    5
   3   11    6    1
   3   12    5    4
   3   13    4    8
   3   13    6    2
   3   15    6    3
   3   17    4   11
   3   17    5    7
   3   17    6    4
   3   18    7    2
   3   19    6    5
   3   21    4   14
   3   21    4   14
   3   21    6    6
   3   21    8    1
   3   22    5   10
   3   23    6    7
   3   25    4   17
   3   25    6    8
   3   25    7    5
   3   27    5   13
   3   27    6    9
   3   27    9    1
   3   29    4   20
   3   29    6   10
   3   29    8    4
   3   30    9    2
   3   31    6   11
3(3+13) = 4(4+8) = 6(6+2), so (3,13,4,8) and (3,13,6,2) are rows.

Cf. A385182, A385592, A385596, A385597.

Clear[solnsM];
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], {#[[1]], #[[2]]} =!= {#[[3]], #[[4]]} &]]]], #[[1]] == m &]];
TableForm[solns = solnsM[3, 140], TableHeadings -> {None, {"m", "u", "v", "w"}}]
aa = Flatten[solns]
Map[#[[2]] &, solns]    (* u, A385595 *)
Map[#[[3]] &, solns]    (* v, A385596 *)
Map[#[[4]] &, solns]    (* w, A385597 *)
(*Peter J.C.Moses, Jun 15 2025*)

A385598 The u sequence in quartets (4,u,v,w); i.e., values of u for solutions to 4(4+u) = v(v+w), in positive integers, v>m, sorted by nondecreasing values of u; see Comments.

Original entry on oeis.org

6, 8, 10, 11, 11, 14, 14, 16, 16, 17, 17, 18, 20, 20, 21, 22, 23, 23, 24, 24, 26, 26, 26, 26, 28, 29, 29, 30, 31, 31, 31, 32, 32, 32, 34, 35, 35, 36, 36, 36, 38, 38, 38, 38, 40, 40, 41, 41, 41, 41, 41, 42, 44, 44, 44, 45, 46, 46, 46, 47, 47, 48, 48, 50, 50

Offset: 1

Clark Kimberling, Jul 10 2025

nonn

A 4-tuple (m,u,v,w) is a quartet if m,u,v,w are positive integers such that m>v and and m*(m+u) = v*(v+w), with the values of u 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=4; for m=1, see A385182.

			First 30 quartets (4,u,v,w):
   m    u    v    w
   4    6    5    3
   4    8    6    2
   4   10    7    1
   4   11    5    7
   4   11    6    4
   4   14    6    6
   4   14    8    1
   4   16    5   11
   4   16    8    2
   4   17    6    8
   4   17    7    5
   4   18    8    3
   4   20    6   10
   4   20    8    4
   4   21    5   15
   4   22    8    5
   4   23    6   12
   4   23    9    3
   4   24    7    9
   4   24    8    6
   4   26    5   19
   4   26    6   14
   4   26    8    7
   4   26   19    2
   4   28    8    8
   4   29    6   16
   4   29   11    1
   4   30    8    9
   4   31    5   23
   4   31    7   13
4(4+16) = 5(5+11) = 8(8+2), so (4,16,5,11) and (4,16,8,2) are rows.

Cf. A385182, A385592, A385599, A385600.

Clear[solnsM];
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], {#[[1]], #[[2]]} =!= {#[[3]], #[[4]]} &]]]], #[[1]] == m &]];
TableForm[solns = solnsM[4, 140], TableHeadings -> {None, {"m", "u", "v", "w"}}]
aa = Flatten[solns]
Map[#[[2]] &, solns]    (* u, A385598 *)
Map[#[[3]] &, solns]    (* v, A385599 *)
Map[#[[4]] &, solns]    (* w, A385600 *)
(*Peter J.C.Moses, Jun 15 2025*)

A386285 Values of u in the quartets (3, u, v, w) of type 2; i.e., values of u for solutions to 3(3 + u) = v(v - w), in positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

Original entry on oeis.org

1, 1, 2, 4, 5, 5, 5, 6, 7, 7, 7, 8, 9, 9, 10, 11, 11, 11, 12, 12, 13, 13, 13, 13, 14, 15, 15, 16, 17, 17, 17, 17, 17, 18, 18, 19, 19, 19, 20, 21, 21, 21, 21, 21, 22, 22, 23, 23, 23, 24, 25, 25, 25, 25, 25, 26, 27, 27, 27, 27, 27, 28, 29, 29, 29, 29, 29, 30

Offset: 1

Clark Kimberling, Aug 12 2025

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 = 3.

			First 20 quartets (3,u,v,w) of type 2:
    m    u    v    w
    3    1    6    4
    3    1   12   11
    3    2   15   14
    3    4   21   20
    3    5    6    2
    3    5   12   10
    3    5   24   23
    3    6   27   26
    3    7    6    1
    3    7   15   13
    3    7   30   29
    3    8   33   32
    3    9   18   16
    3    9   36   35
    3   10   39   38
    3   11    7    1
    3   11   21   19
    3   11   42   41
    3   12    9    4
    3   12   45   44
3(3+2) = 15(15-14), so (3,2,15,14) is in the list.

Cf. A385182 (type 1, m=1), A386286, A386630 (type 3, m=1).

solnsB[t_, u_] := Module[{n = t*(t + u)},
Cases[Select[Divisors[n], # < n/# &],
d_ :> With[{v = n/d, w = n/d - d}, {t, u, v, w} /;
Length[DeleteDuplicates[{t, u, v, w}]] == 4]]];
TableForm[solns = Flatten[Table[Sort[solnsB[3, u]], {u, 50}], 1],
TableHeadings -> {None, {"m", "u", "v", "w"}}]
Map[#[[2]] &, solns] (*u,A386285*)
Map[#[[3]] &, solns] (*v,A386286*)
Map[#[[4]] &, solns] (*w,A386287*)
(* Peter J. C. Moses, Aug 17 2025  *)

A386288 Values of u in the quartets (4, u, v, w) of type 2; i.e., values of u for solutions to 4(4 + u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Aug 12 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 = 4.

			First 20 quartets (4,u,v,w) of type 2:
   m   u    v    w
   4   1   10    8
   4   1   20   19
   4   2    8    5
   4   2   12   10
   4   2   24   23
   4   3   14   12
   4   3   28   27
   4   5   12    9
   4   5   18   16
   4   5   36   35
   4   6    8    3
   4   6   20   18
   4   6   40   39
   4   7   22   20
   4   7   44   43
   4   8   16   13
   4   8   24   22
   4   8   48   47
   4   9   26   24
   4   9   52   51
4(4+2) = 8(8-5), so (4,2,8,5) is in the list.

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

solnsB[t_, u_] := Module[{n = t*(t + u)},
Cases[Select[Divisors[n], # < n/# &],
d_ :> With[{v = n/d, w = n/d - d}, {t, u, v, w} /;
Length[DeleteDuplicates[{t, u, v, w}]] == 4]]];
TableForm[solns = Flatten[Table[Sort[solnsB[4, u]], {u, 26}], 1],
TableHeadings -> {None, {"m", "u", "v", "w"}}]
u1 = Map[#[[2]] &, solns] (*u, A386288 *)
v1 = Map[#[[3]] &, solns] (*v, A386628 *)
w1 = Map[#[[4]] &, solns] (*w, A386629 *)
(* Peter J. C. Moses, Aug 17 2025  *)

cp's OEIS Frontend

A385599 The v sequence in quartets (4,u,v,w); see A385182.

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A385600 The w sequence in quartets (4,u,v,w); see A385182.

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A385183 The v sequence in quartets (1,u,v,w); see A385182.

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A385184 The w sequence in quartets (1,u,v,w); see A385182.

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A385592 Values of u in the quartets (2,u,v,w); i.e., values of u for solutions to 2*(2+u) = v*(v+w), in positive integers, with and v>m, sorted by nondecreasing values of u; see Comments.

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A386218 Values of u in the quartets (1, u, v, w) of type 2; i.e., values of u for solutions to (1 + u) = v(v - w), in positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

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A385595 The u sequence in quartets (3,u,v,w); i.e., values of u for solutions to 3*(3+u) = v*(v+w), in positive integers, with u,v>=3 and u>=m, sorted by nondecreasing values of u; see Comments.

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A385598 The u sequence in quartets (4,u,v,w); i.e., values of u for solutions to 4(4+u) = v(v+w), in positive integers, v>m, sorted by nondecreasing values of u; see Comments.

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A386285 Values of u in the quartets (3, u, v, w) of type 2; i.e., values of u for solutions to 3(3 + u) = v(v - w), in positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

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Mathematica

A386288 Values of u in the quartets (4, u, v, w) of type 2; i.e., values of u for solutions to 4(4 + u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

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Programs

Mathematica

A385592 Values of u in the quartets (2,u,v,w); i.e., values of u for solutions to 2(2+u) = v(v+w), in positive integers, with and v>m, sorted by nondecreasing values of u; see Comments.

A385595 The u sequence in quartets (3,u,v,w); i.e., values of u for solutions to 3(3+u) = v(v+w), in positive integers, with u,v>=3 and u>=m, sorted by nondecreasing values of u; see Comments.