A385882 Values of v in the (1,3)-quartals (m,u,v,w) having m=1; i.e., values of v for solutions to m^1 + u^3 = v^1 + w^3, in positive integers, with m

8, 20, 27, 38, 57, 64, 62, 99, 118, 125, 92, 153, 190, 209, 216, 128, 219, 280, 317, 336, 343, 170, 297, 388, 449, 486, 505, 512, 218, 387, 514, 605, 666, 703, 722, 729, 272, 489, 658, 785, 876, 937, 974, 993, 1000, 332, 603, 820, 989, 1116, 1207, 1268, 1305

Offset: 1

Author

Clark Kimberling, Jul 21 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 (1,u,v,w):
  m   u    v   w
  1   2    8   1
  1   3   20   2
  1   3   27   1
  1   4   38   3
  1   4   57   2
  1   4   64   1
  1   5   62   4
  1   5   99   3
  1   5  118   2
  1   5  125   1
  1   6   92   5
  1   6  153   4
  1   6  190   3
  1   6  209   2
  1   6  216   1
  1   7  128   6
  1   7  219   5
  1   7  280   4
  1   7  317   3
  1   7  336   2
  1   7  343   1
  1   8  170   7
  1   8  297   6
  1   8  388   5
  1   8  449   4
  1   8  486   3
  1   8  505   2
  1   8  512   1
  1   9  218   8
  1   9  387   7
1^1 + 4^3 = 57^1 + 2^3, so (1,4,57,2) is in the list.
		

Crossrefs

Guide to related sequences:

m | u | v | w

--+---------+---------+--------

1 | A003057 | A385882 | A004736

2 | A003057 | A386215 | A004736

3 | A003057 | A386217 | A004736

4 | A003057 | A386219 | A004736

--+---------+---------+---------

Cf. A003057, A004736.

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[1, 1, 3, 2000] (* Solutions restricted to v<2000 *)
Grid[solns]
u1 = Map[#[[2]] &, solns]   (*u, A003057 *)
v1 = Map[#[[3]] &, solns]   (*v, A385882 *)
w1 = Map[#[[4]] &, solns]   (*w, A004736 *)
(* Peter J. C. Moses, Jun 20 2025 *)

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