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A386218 Values of u in the quartets (1, u, v, w) of type 2; 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.

%I A386218 #8 Aug 16 2025 16:04:43
%S A386218 7,9,11,13,14,15,17,17,19,20,21,23,23,23,25,26,27,27,29,29,31,31,32,
%T A386218 33,34,35,35,35,37,38,39,39,39,41,41,43,43,44,44,45,47,47,47,47,49,49,
%U A386218 50,51,51,53,53,53,54,55,55,56,57,59,59,59,59,59,61,62,62
%N A386218 Values of u in the quartets (1, u, v, w) of type 2; 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.
%C A386218 A 4-tuple (m, u, v, w) is a quartet of type 2 if m, u, v, w are distinct positive integers such that m < v and m*(m + u) = v*(v - w). Here, the values of u are arranged 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 = 1.
%e A386218 First 20 quartets (1,u,v,w) of type 2:
%e A386218   m   u   v   w
%e A386218   1   7   4   2
%e A386218   1   9   5   3
%e A386218   1  11   6   4
%e A386218   1  13   7   5
%e A386218   1  14   5   2
%e A386218   1  15   8   6
%e A386218   1  17   6   3
%e A386218   1  17   9   7
%e A386218   1  19  10   8
%e A386218   1  20   7   4
%e A386218   1  21  11   9
%e A386218   1  23   6   2
%e A386218   1  23   8   5
%e A386218   1  23  12  10
%e A386218   1  25  13  11
%e A386218   1  26   9   6
%e A386218   1  27   7   3
%e A386218   1  27  14  12
%e A386218   1  29  10   7
%e A386218   1  29  15  13
%e A386218 Example : 1 (1 + 17) = 6 (6 - 3), so (1, 17, 6, 3) is in the list.
%t A386218 Clear[solnsM];
%t A386218 solnsM[m_, max_] := Module[{ans = {}, rhs = {}, u, v, w, lhs, matching},
%t A386218 Do[Do[AppendTo[rhs, {v*(v - w), v, w}], {w, max}], {v, m*(m + max)}];
%t A386218 rhs = GatherBy[rhs, First];
%t A386218 Do[lhs = m*(m + u); matching = Select[rhs, #[[1, 1]] == lhs &];
%t A386218 If[Length[matching] > 0, Do[AppendTo[ans,
%t A386218 Map[{m, u, #[[2]], #[[3]]} &, matching[[1]]]], {i,
%t A386218 Length[matching]}]], {u, max}];
%t A386218 ans = Flatten[ans, 1];
%t A386218 Select[Union[Map[Sort[{#, RotateLeft[#, 2]}][[1]] &,
%t A386218 Sort[Select[DeleteDuplicates[ans],
%t A386218 Length[Union[#]] == 4 &]]]], #[[1]] == m &]];
%t A386218 TableForm[solns = solnsM[1, 100],
%t A386218 TableHeadings -> {None, {"m", "u", "v", "w"}}]
%t A386218 u1 = Map[#[[2]] &, solns]  (*u, A386218 *)
%t A386218 v1 = Map[#[[3]] &, solns]  (*v, A385883 *)
%t A386218 w1 = Map[#[[4]] &, solns]  (*w, A386630 *)
%t A386218 (* _Peter J. C. Moses_, Jun 15 2025  *)
%Y A386218 Cf. A385182 (type 1), A386630 (type 3).
%K A386218 nonn
%O A386218 1,1
%A A386218 _Clark Kimberling_, Jul 29 2025