cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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.

Original entry on oeis.org

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, 13, 13, 14, 14, 14, 14, 14, 15, 15, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18, 18, 19, 19, 20, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 24
Offset: 1

Views

Author

Clark Kimberling, Aug 12 2025

Keywords

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 = 4.

Examples

			First 20 quartets (4,u,v,w) of type 2:
   m   u    v    w
   4   1   10    8
   4   1   20   19
   4   2    8    5
   4   2   12   10
   4   2   24   23
   4   3   14   12
   4   3   28   27
   4   5   12    9
   4   5   18   16
   4   5   36   35
   4   6    8    3
   4   6   20   18
   4   6   40   39
   4   7   22   20
   4   7   44   43
   4   8   16   13
   4   8   24   22
   4   8   48   47
   4   9   26   24
   4   9   52   51
4(4+2) = 8(8-5), so (4,2,8,5) is in the list.

Crossrefs

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

Programs

  • 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[4, u]], {u, 26}], 1],
TableHeadings -> {None, {"m", "u", "v", "w"}}]
u1 = Map[#[[2]] &, solns] (*u, A386288 *)
v1 = Map[#[[3]] &, solns] (*v, A386628 *)
w1 = Map[#[[4]] &, solns] (*w, A386629 *)
(* Peter J. C. Moses, Aug 17 2025  *)

A386629 Values of w in the (2,3)-quartals (m,u,v,w) having m=1; i.e., values of v for solutions to 1 + u^3 = v^2 + w^3, in positive integers, with mA386627.

Original entry on oeis.org

1, 1, 10, 1, 1, 12, 21, 2, 30, 3, 1, 10, 17, 30, 12, 24, 34, 1, 42, 24, 48, 12, 1, 61, 60, 30, 9, 1, 69, 30, 24, 1, 94, 108, 54, 88, 1, 88, 76, 1, 42, 10, 73, 70, 52, 1, 10, 160, 51, 54, 72, 1, 88, 16, 1, 147, 112, 192, 244, 30, 196, 1, 196, 12, 34, 1, 229
Offset: 1

Views

Author

Clark Kimberling, Jul 29 2025

Keywords

Crossrefs

Cf. A385882, A386215, A386217, A386627, A386628.

A386627 Values of u in the (2,3)-quartals (m,u,v,w) having m=1; i.e., values of v for solutions to 1 + u^3 = v^2 + w^3, in positive integers, with v > 1; see Comments.

Original entry on oeis.org

4, 9, 12, 16, 25, 27, 29, 32, 35, 35, 36, 40, 41, 42, 42, 47, 48, 49, 51, 54, 56, 56, 64, 66, 74, 74, 74, 81, 84, 92, 98, 100, 103, 110, 119, 120, 121, 123, 136, 144, 146, 147, 150, 162, 168, 169, 174, 175, 179, 188, 191, 196, 198, 204, 225, 227, 232, 236
Offset: 1

Views

Author

Clark Kimberling, Jul 28 2025

Keywords

Comments

A 4-tuple (m,u,v,w) is a (p,q)-quartal if m,u,v,w are positive integers such that m
Includes all squares > 1, as 1 + (i^2)^3 = v^2 + w^3 with w = 1, v = i^3. - Robert Israel, Jul 28 2025

Examples

			First 20 (2,3)-quartals (1,u,v,w):
  m    u    v   w
  1    4    8   1
  1    9   27   1
  1   12   27  10
  1   16   64   1
  1   25  125   1
  1   27  134  12
  1   29  123  21
  1   32  181   2
  1   35  126  30
  1   35  207   3
  1   36  216   1
  1   40  251  10
  1   41  253  17
  1   42  217  30
  1   42  269  12
  1   47  300  24
  1   48  267  34
  1   49  343   1
  1   51  242  42
  1   54  379  24
1^2 + 12^3 = 27^2 + 10^3 = 1729, so (1,12,27,10) is in the list.

Crossrefs

Cf. A385882, A386215, A386217, A386628, A386629.

Programs

  • Maple
    f:= proc(u) local t;
  t:= 1+u^3;
  u$nops(select(w -> issqr(t-w^3), [$1 .. u-1]))
end proc:
map(f, [$1..1000]); # Robert Israel, Jul 28 2025
  • Mathematica
    quart[m_, p_, q_, max_] := Module[{ans = {}, lhsD = <||>, lhs, v, u, w, rhs},
   For[u = 1, u <= max, u++, lhs = m^p + u^q;
    AssociateTo[lhsD, lhs -> Append[Lookup[lhsD, lhs, {}], u]];];
   For[v = m + 1, v <= max, v++,
    For[w = 1, w <= max, w++, rhs = v^p + w^q;
      If[KeyExistsQ[lhsD, rhs],
       Do[AppendTo[ans, {m, u, v, w}], {u, lhsD[rhs]}];];];];
   Do[Print["Solution ", i, ": ", ans[[i]], " (", m, "^", p, " + ",
     ans[[i, 2]], "^", q, " = ", ans[[i, 3]], "^", p, " + ",
     ans[[i, 4]], "^", q, " = ", m^p + ans[[i, 2]]^q, ")"], {i,
     Length[ans]}]; ans];
solns = quart[1, 2, 3, 6000]
(* Peter J. C. Moses, Jun 21 2025 *)
Showing 1-3 of 3 results.

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