A386217 Values of v in the (1,3)-quartals (m,u,v,w) having m=3; i.e., values of v for solutions to 3 + u^3 = v + w^3, in positive integers, with m

10, 22, 29, 40, 59, 66, 64, 101, 120, 127, 94, 155, 192, 211, 218, 130, 221, 282, 319, 338, 345, 172, 299, 390, 451, 488, 507, 514, 220, 389, 516, 607, 668, 705, 724, 731, 274, 491, 660, 787, 878, 939, 976, 995, 1002, 334, 605, 822, 991, 1118, 1209, 1270, 1307

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

Examples

			First 20 (1,3)-quartals (3,u,v,w):
   m   u    v   w
   3   2   10   1
   3   3   22   2
   3   3   29   1
   3   4   40   3
   3   4   59   2
   3   4   66   1
   3   5   64   4
   3   5  101   3
   3   5  120   2
   3   5  127   1
   3   6   94   5
   3   6  155   4
   3   6  192   3
   3   6  211   2
   3   6  218   1
   3   7  130   6
   3   7  221   5
   3   7  282   4
   3   7  319   3
   3   7  338   2
3^1 + 4^3 = 40^1 + 3^3, so (3,4,40,3) is in the list.
		

Crossrefs

Cf. A385882, A386215, A386219.

Programs

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[3, 1, 3, 2000]
Grid[solns]
(* Peter J. C. Moses, Jun 21 2025 *)

Formula

As a triangle T(u,k), 1 <= k <= u-1, T(u,k) = 3+u^3-(u-k)^3. - Pontus von Brömssen, Aug 03 2025

a(n) = A385882(n)+2 = A386215(n)+1 = A386219(n)-1. - Pontus von Brömssen, Aug 04 2025

Extensions

Data corrected by Sean A. Irvine, Aug 01 2025

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