A386219 -id:A386219 - cp's OEIS frontend

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

Clark Kimberling, Jul 21 2025

nonn

A 4-tuple (m,u,v,w) is a (p,q)-quartal if m,u,v,w are positive integers such that m

			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.

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.

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 *)

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

Original entry on oeis.org

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

Clark Kimberling, Jul 28 2025

nonn

A 4-tuple (m,u,v,w) is a (p,q)-quartal if m,u,v,w are positive integers such that m

			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.

Cf. A385882, A386215, A386219.

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 *)

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

Data corrected by Sean A. Irvine, Aug 01 2025

Showing 1-2 of 2 results.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica