cp's OEIS Frontend

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.

%I A385592 #19 Jul 11 2025 01:13:21
%S A385592 4,7,8,10,10,12,13,13,14,16,16,18,18,19,19,20,22,22,22,23,24,25,25,26,
%T A385592 26,28,28,28,28,30,31,31,32,33,33,34,34,34,34,36,37,37,38,38,38,40,40,
%U A385592 40,40,42,42,43,43,43,43,44,46,46,46,46,47,48,48,49,49
%N 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.
%C A385592 A 4-tuple (m,u,v,w) is a quartet if m,u,v,w are positive integers such that m<v 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=2; for m=1, see A385182.
%e A385592 First 30 quartets (2,u,v,w):
%e A385592   m    u   v   w
%e A385592   2    4   3   1
%e A385592   2    7   3   3
%e A385592   2    8   4   1
%e A385592   2   10   3   5
%e A385592   2   10   4   2
%e A385592   2   12   4   3
%e A385592   2   13   3   7
%e A385592   2   13   5   1
%e A385592   2   14   4   4
%e A385592   2   16   3   9
%e A385592   2   16   4   5
%e A385592   2   18   4   6
%e A385592   2   18   5   3
%e A385592   2   19   3  11
%e A385592   2   19   6   1
%e A385592   2   20   4   7
%e A385592   2   22   3  13
%e A385592   2   22   4   8
%e A385592   2   22   6   2
%e A385592   2   23   5   5
%e A385592   2   24   4   9
%e A385592   2   25   3  15
%e A385592   2   25   6   3
%e A385592   2   26   4  10
%e A385592   2   26   7   1
%e A385592   2   28   3  17
%e A385592   2   28   4  11
%e A385592   2   28   5   7
%e A385592   2   28   6   4
%e A385592   2   30   4  12
%e A385592 2(2+16) = 3(3+9) = 4(4+5), so (2,16,3,9) and (2,16,4,5) are rows.
%t A385592 Clear[solnsM];
%t A385592 solnsM[m_, max_] := Module[{ans = {}, rhs = {}, u, v, w, lhs, matching},
%t A385592 Do[Do[AppendTo[rhs, {v*(v + w), v, w}], {w, max}], {v, m*(m + max)}];
%t A385592 rhs = GatherBy[rhs, First];
%t A385592 Do[lhs = m*(m + u); matching = Select[rhs, #[[1, 1]] == lhs &];
%t A385592 If[Length[matching] > 0, Do[AppendTo[ans,
%t A385592 Map[{m, u, #[[2]], #[[3]]} &, matching[[1]]]], {i,
%t A385592 Length[matching]}]], {u, max}];
%t A385592 ans = Flatten[ans, 1];
%t A385592 Select[Union[Map[Sort[{#, RotateLeft[#, 2]}][[1]] &,
%t A385592 Sort[Select[DeleteDuplicates[
%t A385592 ans], {#[[1]], #[[2]]} =!= {#[[3]], #[[4]]} &]]]], #[[1]] == m &]];
%t A385592 TableForm[solns = solnsM[2, 140], TableHeadings -> {None, {"m", "u", "v", "w"}}]
%t A385592 aa = Flatten[solns]
%t A385592 Map[#[[2]] &, solns]    (* u, A385592 *)
%t A385592 Map[#[[3]] &, solns]    (* v, A385593 *)
%t A385592 Map[#[[4]] &, solns]    (* w, A385594 *)
%t A385592 (* _Peter J. C. Moses_, Jun 15 2025 *)
%Y A385592 Cf. A385182, A385592, A385593, A385594.
%K A385592 nonn
%O A385592 1,1
%A A385592 _Clark Kimberling_, Jul 04 2025