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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.

%I A385598 #5 Jul 12 2025 18:53:22
%S A385598 6,8,10,11,11,14,14,16,16,17,17,18,20,20,21,22,23,23,24,24,26,26,26,
%T A385598 26,28,29,29,30,31,31,31,32,32,32,34,35,35,36,36,36,38,38,38,38,40,40,
%U A385598 41,41,41,41,41,42,44,44,44,45,46,46,46,47,47,48,48,50,50
%N 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.
%C A385598 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.
%e A385598 First 30 quartets (4,u,v,w):
%e A385598    m    u    v    w
%e A385598    4    6    5    3
%e A385598    4    8    6    2
%e A385598    4   10    7    1
%e A385598    4   11    5    7
%e A385598    4   11    6    4
%e A385598    4   14    6    6
%e A385598    4   14    8    1
%e A385598    4   16    5   11
%e A385598    4   16    8    2
%e A385598    4   17    6    8
%e A385598    4   17    7    5
%e A385598    4   18    8    3
%e A385598    4   20    6   10
%e A385598    4   20    8    4
%e A385598    4   21    5   15
%e A385598    4   22    8    5
%e A385598    4   23    6   12
%e A385598    4   23    9    3
%e A385598    4   24    7    9
%e A385598    4   24    8    6
%e A385598    4   26    5   19
%e A385598    4   26    6   14
%e A385598    4   26    8    7
%e A385598    4   26   19    2
%e A385598    4   28    8    8
%e A385598    4   29    6   16
%e A385598    4   29   11    1
%e A385598    4   30    8    9
%e A385598    4   31    5   23
%e A385598    4   31    7   13
%e A385598 4(4+16) = 5(5+11) = 8(8+2), so (4,16,5,11) and (4,16,8,2) are rows.
%t A385598 Clear[solnsM];
%t A385598 solnsM[m_, max_] := Module[{ans = {}, rhs = {}, u, v, w, lhs, matching},
%t A385598 Do[Do[AppendTo[rhs, {v*(v + w), v, w}], {w, max}], {v, m*(m + max)}];
%t A385598 rhs = GatherBy[rhs, First];
%t A385598 Do[lhs = m*(m + u); matching = Select[rhs, #[[1, 1]] == lhs &];
%t A385598 If[Length[matching] > 0, Do[AppendTo[ans,
%t A385598 Map[{m, u, #[[2]], #[[3]]} &, matching[[1]]]], {i,
%t A385598 Length[matching]}]], {u, max}];
%t A385598 ans = Flatten[ans, 1];
%t A385598 Select[Union[Map[Sort[{#, RotateLeft[#, 2]}][[1]] &,
%t A385598 Sort[Select[DeleteDuplicates[
%t A385598 ans], {#[[1]], #[[2]]} =!= {#[[3]], #[[4]]} &]]]], #[[1]] == m &]];
%t A385598 TableForm[solns = solnsM[4, 140], TableHeadings -> {None, {"m", "u", "v", "w"}}]
%t A385598 aa = Flatten[solns]
%t A385598 Map[#[[2]] &, solns]    (* u, A385598 *)
%t A385598 Map[#[[3]] &, solns]    (* v, A385599 *)
%t A385598 Map[#[[4]] &, solns]    (* w, A385600 *)
%t A385598 (*_Peter J.C.Moses_, Jun 15 2025*)
%Y A385598 Cf. A385182, A385592, A385599, A385600.
%K A385598 nonn
%O A385598 1,1
%A A385598 _Clark Kimberling_, Jul 10 2025