A385598 -id:A385598 - cp's OEIS frontend

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.

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

Clark Kimberling, Jun 23 2025

nonn

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

			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).

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.

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*)

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

Original entry on oeis.org

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.

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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.

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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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