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A386631 Values of u in the quartets (2, u, v, w) of type 3; i.e., values of u for solutions to 2(2 - u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

%I A386631 #6 Aug 26 2025 23:27:30
%S A386631 5,6,7,8,8,8,9,10,10,11,11,11,12,12,12,13,14,14,14,14,14,15,16,16,16,
%T A386631 17,17,17,17,17,18,18,18,19,20,20,20,20,20,20,21,22,22,22,22,22,23,23,
%U A386631 23,23,23,24,24,24,25,26,26,26,26,26,26,26,27,27,27,28
%N A386631 Values of u in the quartets (2, u, v, w) of type 3; i.e., values of u for solutions to 2(2 - u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.
%C A386631 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 = 2.
%e A386631 First 20 quartets (2,u,v,w) of type 3:
%e A386631    m    u    v    w
%e A386631    2    5    6    7
%e A386631    2    6    8    9
%e A386631    2    7   10   11
%e A386631    2    8    3    7
%e A386631    2    8    4    7
%e A386631    2    8   12   13
%e A386631    2    9   14   15
%e A386631    2   10    4    8
%e A386631    2   10   16   17
%e A386631    2   11    3    9
%e A386631    2   11    6    9
%e A386631    2   11   18   19
%e A386631    2   12    4    9
%e A386631    2   12    5    9
%e A386631    2   12   20   21
%e A386631    2   13   22   23
%e A386631    2   14    3   11
%e A386631    2   14    4   10
%e A386631    2   14    6   10
%e A386631    2   14    8   11
%e A386631 2(2-10) = 4(4-8), so (2, 10, 4, 8) is in the list.
%t A386631 ssolnsM[m_Integer?Positive, u_Integer?Positive] :=
%t A386631   Module[{n = m  (m - u), nn, sgn, ds, tups}, If[n == 0, Return[{}]];
%t A386631    sgn = Sign[n]; nn = Abs[n];
%t A386631    ds = Divisors[nn];
%t A386631    If[sgn > 0, ds = Select[ds, # < nn/# &]];
%t A386631    tups = ({m, u, nn/#, nn/# - sgn  #} & /@ ds);
%t A386631    Select[tups, #[[3]] > 1 && #[[4]] > 0 && #[[2]] =!= #[[4]] &&
%t A386631    Length@DeleteDuplicates[#] == 4 &]];
%t A386631 (solns = Sort[Flatten[Map[solnsM[2, #] &, Range[2, 60]], 1]]) // ColumnForm
%t A386631 Map[#[[2]] &, solns] (*A386631*)
%t A386631 Map[#[[3]] &, solns] (*A387225*)
%t A386631 Map[#[[4]] &, solns] (*A387226*)
%t A386631 (* _Peter J. C. Moses_, Aug 22 2025 *)
%Y A386631 Cf. A385182 (type 1), A386218 (type 2), A385476 (type 3, m=1), A387225, A387226.
%K A386631 nonn,new
%O A386631 1,1
%A A386631 _Clark Kimberling_, Aug 22 2025