cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Clark Kimberling, Jul 04 2025

Keywords

Crossrefs

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

Views

Author

Clark Kimberling, Jul 07 2025

Keywords

Crossrefs

Cf. A385182, A385592, A385593.

A385182 Values of u in the quartets (1,u,v,w); 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

5, 7, 9, 11, 11, 13, 14, 15, 17, 17, 19, 19, 20, 21, 23, 23, 23, 25, 26, 27, 27, 29, 29, 29, 31, 31, 32, 33, 34, 35, 35, 35, 37, 38, 39, 39, 39, 41, 41, 41, 43, 43, 44, 44, 45, 47, 47, 47, 47, 49, 49, 50, 51, 51, 53, 53, 53, 54, 55, 55, 55, 56, 57, 59, 59
Offset: 1

Views

Author

Clark Kimberling, Jun 23 2025

Keywords

Comments

A 4-tuple (m,u,v,w) is a quartet if m,u,v,w are positive integers such that m
If m is a prime, then (u,v,w) = (m+2,m+1,m-1) is the first solution (in the defined ordering of triples).
u >= 1 appears A056924(u+1)-1 times. - Pontus von Brömssen, Jul 06 2025

Examples

			First 30 quartets (1,u,v,w):
  m    u    v    w
  1    5    2    1
  1    7    2    2
  1    9    2    3
  1   11    2    4
  1   11    3    1
  1   13    2    5
  1   14    3    2
  1   15    2    6
  1   17    2    7
  1   17    3    3
  1   19    2    8
  1   19    4    1
  1   20    3    4
  1   21    2    9
  1   23    2   10
  1   23    3    5
  1   23    4    2
  1   25    2   11
  1   26    3    6
  1   27    2   12
  1   27    4    3
  1   29    2   13
  1   29    3    7
  1   29    5    1
  1   31    2   14
  1   31    4    4
  1   32    3    8
  1   33    2   15
  1   34    5    2
  1   35    2   16
1*(1+23) = 2*(2+10) = 3*(3+5) = 4*(4+2), so three of the rows are (1,23,2,10), (1,23,3,5), and (1,23,4,2).

Crossrefs

Guide to related sequences:
m | u | v | w
--+---------+---------+--------
1 | A385182 | A385183 | A385184
2 | A385592 | A385593 | A385594
3 | A385595 | A385596 | A385597
4 | A385598 | A385599 | A385600
--+---------+---------+--------
Cf. A056924.

Programs

  • Mathematica
    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[1, 140], TableHeadings -> {None, {"m", "u", "v", "w"}}]
aa = Flatten[solns]
Map[#[[2]] &, solns]    (* u, A385182 *)
Map[#[[3]] &, solns]    (* v, A385183 *)
Map[#[[4]] &, solns]    (* w, A385184 *)
(*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

Views

Author

Clark Kimberling, Jul 07 2025

Keywords

Comments

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

Examples

			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.

Crossrefs

Cf. A385182, A385592, A385596, A385597.

Programs

  • Mathematica
    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

Views

Author

Clark Kimberling, Jul 10 2025

Keywords

Comments

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.

Examples

			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.

Crossrefs

Cf. A385182, A385592, A385599, A385600.

Programs

  • Mathematica
    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*)
Showing 1-5 of 5 results.

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