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
Keywords
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.
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
Keywords
Comments
A 4-tuple (m,u,v,w) is a quartet if m,u,v,w are positive integers such that mA385182.
Examples
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.
Programs
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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[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 *)
A385594 The w sequence in quartets (2,u,v,w); see A385592.
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
Comments
Examples
Crossrefs
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*)