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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.
4
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, 16, 17, 17, 17, 17, 17, 18, 18, 19, 19, 19, 20, 21, 21, 21, 21, 21, 22, 22, 23, 23, 23, 24, 25, 25, 25, 25, 25, 26, 27, 27, 27, 27, 27, 28, 29, 29, 29, 29, 29, 30
OFFSET
1,3
COMMENTS
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.
EXAMPLE
First 20 quartets (3,u,v,w) of type 2:
m u v w
3 1 6 4
3 1 12 11
3 2 15 14
3 4 21 20
3 5 6 2
3 5 12 10
3 5 24 23
3 6 27 26
3 7 6 1
3 7 15 13
3 7 30 29
3 8 33 32
3 9 18 16
3 9 36 35
3 10 39 38
3 11 7 1
3 11 21 19
3 11 42 41
3 12 9 4
3 12 45 44
3(3+2) = 15(15-14), so (3,2,15,14) is in the list.
MATHEMATICA
solnsB[t_, u_] := Module[{n = t*(t + u)},
Cases[Select[Divisors[n], # < n/# &],
d_ :> With[{v = n/d, w = n/d - d}, {t, u, v, w} /;
Length[DeleteDuplicates[{t, u, v, w}]] == 4]]];
TableForm[solns = Flatten[Table[Sort[solnsB[3, u]], {u, 50}], 1],
TableHeadings -> {None, {"m", "u", "v", "w"}}]
Map[#[[2]] &, solns] (*u, A386285*)
Map[#[[3]] &, solns] (*v, A386286*)
Map[#[[4]] &, solns] (*w, A386287*)
(* Peter J. C. Moses, Aug 17 2025 *)
CROSSREFS
Cf. A385182 (type 1, m=1), A386286, A386630 (type 3, m=1).
Sequence in context: A034214 A317749 A253415 * A227401 A380110 A131813
KEYWORD
nonn
AUTHOR
Clark Kimberling, Aug 12 2025
STATUS
approved