A386215 Values of v in the (1,3)-quartals (m,u,v,w) having m=2; i.e., values of v for solutions to m + u^3 = v + w^3, in positive integers, with m

9, 21, 28, 39, 58, 65, 63, 100, 119, 126, 93, 154, 191, 210, 217, 129, 220, 281, 318, 337, 344, 171, 298, 389, 450, 487, 506, 513, 219, 388, 515, 606, 667, 704, 723, 730, 273, 490, 659, 786, 877, 938, 975, 994, 1001, 333, 604, 821, 990, 1117, 1208, 1269

Offset: 1

Author

Clark Kimberling, Jul 22 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

Examples

			First thirty (1,3)-quartals (2,u,v,w):
  m     u      v     w
  2     2      9     1
  2     3     21     2
  2     3     28     1
  2     4     39     3
  2     4     58     2
  2     4     65     1
  2     5     63     4
  2     5    100     3
  2     5    119     2
  2     5    126     1
  2     6     93     5
  2     6    154     4
  2     6    191     3
  2     6    210     2
  2     6    217     1
  2     7    129     6
  2     7    220     5
  2     7    281     4
  2     7    318     3
  2     7    337     2
  2     7    344     1
  2     8    171     7
  2     8    298     6
  2     8    389     5
  2     8    450     4
  2     8    487     3
  2     8    506     2
  2     8    513     1
  2     9    219     8
  2     9    388     7
2^1 + 3^3 = 21^1 + 2^3, so (2,3,21,2) is in the list.
		

Crossrefs

Cf. A003057, A004736, A385882.

Programs

Maple

seq(seq(2 + u^3 - w^3, w = u-1 .. 1,-1),u=2..20); # Robert Israel, Jul 27 2025

Mathematica

quartals[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]}];];];];
   ans = SortBy[ans, #[[2]] &];
   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 = quartals[2, 1, 3, 2000]  (* solutions restricted to v<2000 *)
Grid[solns]
u1 = Map[#[[2]] &, solns]  (*u, A003057 *)
v1 = Map[#[[3]] &, solns]  (*v, A386215 *)
w1 = Map[#[[4]] &, solns]  (*w, A004736 *)
(* Peter J. C. Moses, Jun 20 2025 *)

Formula

a(n) = 2 + u^3 - (u*(u-1)/2 + 1 - n)^3 where u = floor((3+sqrt(8*n-7))/2). - Robert Israel, Jul 27 2025

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