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A386288 Values of u in the quartets (4, u, v, w) of type 2; 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.

%I A386288 #12 Aug 19 2025 16:41:32
%S A386288 1,1,2,2,2,3,3,5,5,5,6,6,6,7,7,8,8,8,9,9,10,10,10,11,11,11,11,12,12,
%T A386288 13,13,14,14,14,14,14,15,15,16,16,16,17,17,17,17,17,18,18,18,19,19,20,
%U A386288 20,20,20,21,21,21,22,22,22,23,23,23,23,23,24,24,24,24
%N A386288 Values of u in the quartets (4, u, v, w) of type 2; 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 A386288 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 = 4.
%e A386288 First 20 quartets (4,u,v,w) of type 2:
%e A386288    m   u    v    w
%e A386288    4   1   10    8
%e A386288    4   1   20   19
%e A386288    4   2    8    5
%e A386288    4   2   12   10
%e A386288    4   2   24   23
%e A386288    4   3   14   12
%e A386288    4   3   28   27
%e A386288    4   5   12    9
%e A386288    4   5   18   16
%e A386288    4   5   36   35
%e A386288    4   6    8    3
%e A386288    4   6   20   18
%e A386288    4   6   40   39
%e A386288    4   7   22   20
%e A386288    4   7   44   43
%e A386288    4   8   16   13
%e A386288    4   8   24   22
%e A386288    4   8   48   47
%e A386288    4   9   26   24
%e A386288    4   9   52   51
%e A386288 4(4+2) = 8(8-5), so (4,2,8,5) is in the list.
%t A386288 solnsB[t_, u_] := Module[{n = t*(t + u)},
%t A386288 Cases[Select[Divisors[n], # < n/# &],
%t A386288 d_ :> With[{v = n/d, w = n/d - d}, {t, u, v, w} /;
%t A386288 Length[DeleteDuplicates[{t, u, v, w}]] == 4]]];
%t A386288 TableForm[solns = Flatten[Table[Sort[solnsB[4, u]], {u, 26}], 1],
%t A386288 TableHeadings -> {None, {"m", "u", "v", "w"}}]
%t A386288 u1 = Map[#[[2]] &, solns] (*u, A386288 *)
%t A386288 v1 = Map[#[[3]] &, solns] (*v, A386628 *)
%t A386288 w1 = Map[#[[4]] &, solns] (*w, A386629 *)
%t A386288 (* _Peter J. C. Moses_, Aug 17 2025  *)
%Y A386288 Cf. A385182 (type 1, m=1), A386630 (type 3, m=1).
%K A386288 nonn
%O A386288 1,3
%A A386288 _Clark Kimberling_, Aug 12 2025