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A385476 Values of u in the quartets (1, u, v, w) of type 3; i.e., values of u for solutions to (1 - u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

%I A385476 #28 Aug 26 2025 23:38:09
%S A385476 5,7,7,9,9,10,11,11,13,13,13,13,15,15,16,16,17,17,17,19,19,19,19,21,
%T A385476 21,21,21,22,22,23,23,25,25,25,25,25,25,26,27,27,28,28,29,29,29,29,31,
%U A385476 31,31,31,31,31,33,33,33,33,34,34,35,35,36,36,37,37,37,37
%N A385476 Values of u in the quartets (1, u, v, w) of type 3; i.e., values of u for solutions to (1 - u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.
%C A385476 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 = 1.
%e A385476 First 20 quartets (1,u,v,w) of type 3:
%e A385476    m    u    v    w
%e A385476    1    5    2    4
%e A385476    1    7    2    5
%e A385476    1    7    3    5
%e A385476    1    9    2    6
%e A385476    1    9    4    6
%e A385476    1   10    3    6
%e A385476    1   11    2    7
%e A385476    1   11    5    7
%e A385476    1   13    2    8
%e A385476    1   13    3    7
%e A385476    1   13    4    7
%e A385476    1   13    6    8
%e A385476    1   15    2    9
%e A385476    1   15    7    9
%e A385476    1   16    3    5
%e A385476    1   16    3    8
%e A385476    1   17    2   10
%e A385476    1   17    4    8
%e A385476    1   17    8   10
%e A385476    1   19    2   11
%e A385476 1(1-11) = 5(5-7), so (1, 11, 5, 7) is in the list.
%t A385476 solnsM[m_Integer?Positive, u_Integer?Positive] :=
%t A385476   Module[{n = m  (m - u), nn, sgn, ds, tups}, If[n == 0, Return[{}]];
%t A385476    sgn = Sign[n]; nn = Abs[n];
%t A385476    ds = Divisors[nn];
%t A385476    If[sgn > 0, ds = Select[ds, # < nn/# &]];
%t A385476    tups = ({m, u, nn/#, nn/# - sgn  #} & /@ ds);
%t A385476    Select[tups, #[[3]] > 1 && #[[4]] > 0 && #[[2]] =!= #[[4]](*&&
%t A385476      Length@DeleteDuplicates[#]==4*)&]];
%t A385476 (solns =
%t A385476    Sort[Flatten[Map[solnsM[1, #] &, Range[2, 30]], 1]]) // ColumnForm
%t A385476 Map[#[[2]] &, solns] (*A385476*)
%t A385476 Map[#[[3]] &, solns] (*A163870*)
%t A385476 Map[#[[4]] &, solns] (*A385246*)
%t A385476 (* _Peter J. C. Moses_, Aug 22 2025 *)
%Y A385476 Cf. A385182 (type 1), A386218 (type 2), A386631, A385246.
%K A385476 nonn,new
%O A385476 1,1
%A A385476 _Clark Kimberling_, Aug 16 2025