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A386285 Values of u in the quartets (3, u, v, w) of type 2; i.e., values of u for solutions to 3(3 + u) = v(v - w), in positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

%I A386285 #9 Aug 24 2025 18:37:36
%S A386285 1,1,2,4,5,5,5,6,7,7,7,8,9,9,10,11,11,11,12,12,13,13,13,13,14,15,15,
%T A386285 16,17,17,17,17,17,18,18,19,19,19,20,21,21,21,21,21,22,22,23,23,23,24,
%U A386285 25,25,25,25,25,26,27,27,27,27,27,28,29,29,29,29,29,30
%N A386285 Values of u in the quartets (3, u, v, w) of type 2; i.e., values of u for solutions to 3(3 + u) =  v(v - w), in positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.
%C A386285 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 = 3.
%e A386285 First 20 quartets (3,u,v,w) of type 2:
%e A386285     m    u    v    w
%e A386285     3    1    6    4
%e A386285     3    1   12   11
%e A386285     3    2   15   14
%e A386285     3    4   21   20
%e A386285     3    5    6    2
%e A386285     3    5   12   10
%e A386285     3    5   24   23
%e A386285     3    6   27   26
%e A386285     3    7    6    1
%e A386285     3    7   15   13
%e A386285     3    7   30   29
%e A386285     3    8   33   32
%e A386285     3    9   18   16
%e A386285     3    9   36   35
%e A386285     3   10   39   38
%e A386285     3   11    7    1
%e A386285     3   11   21   19
%e A386285     3   11   42   41
%e A386285     3   12    9    4
%e A386285     3   12   45   44
%e A386285 3(3+2) = 15(15-14), so (3,2,15,14) is in the list.
%t A386285 solnsB[t_, u_] := Module[{n = t*(t + u)},
%t A386285 Cases[Select[Divisors[n], # < n/# &],
%t A386285 d_ :> With[{v = n/d, w = n/d - d}, {t, u, v, w} /;
%t A386285 Length[DeleteDuplicates[{t, u, v, w}]] == 4]]];
%t A386285 TableForm[solns = Flatten[Table[Sort[solnsB[3, u]], {u, 50}], 1],
%t A386285 TableHeadings -> {None, {"m", "u", "v", "w"}}]
%t A386285 Map[#[[2]] &, solns] (*u,A386285*)
%t A386285 Map[#[[3]] &, solns] (*v,A386286*)
%t A386285 Map[#[[4]] &, solns] (*w,A386287*)
%t A386285 (* _Peter J. C. Moses_, Aug 17 2025  *)
%Y A386285 Cf. A385182 (type 1, m=1), A386286, A386630 (type 3, m=1).
%K A386285 nonn,changed
%O A386285 1,3
%A A386285 _Clark Kimberling_, Aug 12 2025