%I #6 Nov 01 2025 14:56:04

%S 1,2,6,7,7,8,9,9,10,10,10,11,11,12,12,13,13,13,13,14,14,14,14,15,15,

%T 16,16,16,16,17,17,18,18,18,18,19,19,19,19,19,19,19,20,20,20,21,21,22,

%U 22,22,22,22,22,22,23,23,24,24,24,24,24,24,25,25,25,25,25

%N Values of u in the quartets (4, u, v, w) of type 3; i.e., values of u for solutions to 4*(4 - u) = v*(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

%C A 4-tuple (m, u, v, w) is a quartet of type 3 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 = 4.

%e First 20 quartets (4,u,v,w) of type 3:

%e m u v w

%e 4 1 12 11

%e 4 2 8 7

%e 4 6 8 9

%e 4 7 6 8

%e 4 7 12 13

%e 4 8 16 17

%e 4 9 10 12

%e 4 9 20 21

%e 4 10 8 11

%e 4 10 12 14

%e 4 10 24 25

%e 4 11 14 16

%e 4 11 28 29

%e 4 12 16 18

%e 4 12 32 33

%e 4 13 6 12

%e 4 13 12 15

%e 4 13 18 20

%e 4 13 36 37

%e 4 14 5 13

%e 4*(4-7) = 6*(6-8), so (4,7,6,8) is in the list.

%t solnsM[m_, max_] := Module[{ans = {}, rhs = {}, u, v, w, lhs, matching},

%t Do[Do[AppendTo[rhs, {v*(v - w), v, w}], {w, max}], {v, m*(m + max)}];

%t rhs = GatherBy[rhs, First];

%t Do[lhs = m*(m - u); matching = Select[rhs, #[[1, 1]] == lhs &];

%t If[Length[matching] > 0, Do[AppendTo[ans,

%t Map[{m, u, #[[2]], #[[3]]} &, matching[[1]]]], {i,

%t Length[matching]}]], {u, max}];

%t ans = Flatten[ans, 1];

%t Select[Union[Map[Sort[{#, RotateLeft[#, 2]}][[1]] &,

%t Sort[Select[DeleteDuplicates[ans],

%t Length[Union[#]] == 4 &]]]], #[[1]] == m &]];

%t TableForm[solns = solnsM[4, 100],

%t TableHeadings -> {None, {"m", "u", "v", "w"}}]

%t Map[#[[2]] &, solns] (* u, A390024 *)

%t Map[#[[3]] &, solns] (* v, A390025 *)

%t Map[#[[4]] &, solns] (* w, A390026 *)

%t (* _Peter J. C. Moses_, Jun 15 2025 *)

%Y Cf. A385182 (type 1), A386218 (type 2), A386630, A385476, A390025, A390026.

%K nonn

%O 1,2

%A _Clark Kimberling_, Oct 29 2025