cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 21 2025

Keywords

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 *)

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

Views

Author

Clark Kimberling, Jul 28 2025

Keywords

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

A386628 Values of v 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 mA386217.

Original entry on oeis.org

8, 27, 27, 64, 125, 134, 123, 181, 126, 207, 216, 251, 253, 217, 269, 300, 267, 343, 242, 379, 255, 417, 512, 246, 435, 615, 636, 729, 514, 867, 963, 1000, 512, 267, 1236, 1023, 1331, 1086, 1441, 1728, 1743, 1782, 1728, 1977, 2145, 2197, 2295, 1124, 2367
Offset: 1

Views

Author

Clark Kimberling, Jul 29 2025

Keywords

Crossrefs

Cf. A385882, A386215, A386217, A386627, A386629.

A386629 Values of w 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 mA386627.

Original entry on oeis.org

1, 1, 10, 1, 1, 12, 21, 2, 30, 3, 1, 10, 17, 30, 12, 24, 34, 1, 42, 24, 48, 12, 1, 61, 60, 30, 9, 1, 69, 30, 24, 1, 94, 108, 54, 88, 1, 88, 76, 1, 42, 10, 73, 70, 52, 1, 10, 160, 51, 54, 72, 1, 88, 16, 1, 147, 112, 192, 244, 30, 196, 1, 196, 12, 34, 1, 229
Offset: 1

Views

Author

Clark Kimberling, Jul 29 2025

Keywords

Crossrefs

Cf. A385882, A386215, A386217, A386627, A386628.

A386219 Values of v in the (1,3)-quartals (m,u,v,w) having m=4; i.e., values of v for solutions to 4 + u^3 = v + w^3, in positive integers, with m

Original entry on oeis.org

11, 23, 30, 41, 60, 67, 65, 102, 121, 128, 95, 156, 193, 212, 219, 131, 222, 283, 320, 339, 346, 173, 300, 391, 452, 489, 508, 515, 221, 390, 517, 608, 669, 706, 725, 732, 275, 492, 661, 788, 879, 940, 977, 996, 1003, 335, 606, 823, 992, 1119, 1210, 1271, 1308
Offset: 1

Views

Author

Clark Kimberling, Jul 28 2025

Keywords

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 (4,u,v,w):
  m   u    v   w
  4   2   11   1
  4   3   23   2
  4   3   30   1
  4   4   41   3
  4   4   60   2
  4   4   67   1
  4   5   65   4
  4   5  102   3
  4   5  121   2
  4   5  128   1
  4   6   95   5
  4   6  156   4
  4   6  193   3
  4   6  212   2
  4   6  219   1
  4   7  131   6
  4   7  222   5
  4   7  283   4
  4   7  320   3
  4   7  339   2
4^1 + 5^3 = 65^1 + 4^3, so (4,5,65,4) is in the list.

Crossrefs

Cf. A385882, A386215, A386217.

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[4, 1, 3, 2000]
Grid[solns]
(* Peter J. C. Moses, Jun 20 2025 *)

Extensions

Data corrected by Sean A. Irvine, Aug 11 2025

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.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 28 2025

Keywords

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 *)
Showing 1-6 of 6 results.

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