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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<v, sorted by u, then v.
7
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
1,1
COMMENTS
A 4-tuple (m,u,v,w) is a (p,q)-quartal if m,u,v,w are positive integers such that m<v and m^p + u^q = v^p + w^q. Here, m = 2, p = 1, q = 3.
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
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 22 2025
STATUS
approved