A386286 -id:A386286 - cp's OEIS frontend

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.

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

Clark Kimberling, Aug 12 2025

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.

			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.

Cf. A385182 (type 1, m=1), A386286, A386630 (type 3, m=1).

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  *)

A385879 Values of u in triples (u, v, w) such that the polynomial x^3 + ux^2 + vx + w has 3 (possibly repeated) negative integer zeros; the triples are ordered by the inequality u < v.

Original entry on oeis.org

3, 4, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14

Offset: 1

Clark Kimberling, Jul 11 2025

nonn

			First 20 triples:
    u    v    w
    3    3    1
    4    5    2
    5    7    3
    5    8    4
    6    9    4
    6   11    6
    6   12    8
    7   11    5
    7   14    8
    7   15    9
    7   16   12
    8   13    6
    8   17   10
    8   19   12
    8   20   16
    8   21   18
    9   15    7
    9   20   12
    9   23   15
    9   24   16
(x + 1)^3 = x^3 + 3*x^2 + 3*x + 1, so (3, 3, 1) is in the list; here the negative zeros are -1, -1, and -1.

Cf. A385880, A386285, A386286.

z = 120;
t = Table[{b + c + d, c  d + d  b + b  c, b  c  d}, {b, 1, z}, {c, 1, z}, {d, 1, z}];
t1 = Union[Flatten[t, 2]]; t2 = Take[t1, 40]
Grid[t2]

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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.

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A385879 Values of u in triples (u, v, w) such that the polynomial x^3 + ux^2 + vx + w has 3 (possibly repeated) negative integer zeros; the triples are ordered by the inequality u < v.

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cp's OEIS Frontend

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.

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Mathematica

A385879 Values of u in triples (u, v, w) such that the polynomial x^3 + u*x^2 + v*x + w has 3 (possibly repeated) negative integer zeros; the triples are ordered by the inequality u < v.

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Mathematica

A385879 Values of u in triples (u, v, w) such that the polynomial x^3 + ux^2 + vx + w has 3 (possibly repeated) negative integer zeros; the triples are ordered by the inequality u < v.