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.

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

Author

Clark Kimberling, Jul 28 2025

Keywords

• nonn

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 *)

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