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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 sorted by u, then v.

%I A386215 #11 Jul 27 2025 16:30:19
%S A386215 9,21,28,39,58,65,63,100,119,126,93,154,191,210,217,129,220,281,318,
%T A386215 337,344,171,298,389,450,487,506,513,219,388,515,606,667,704,723,730,
%U A386215 273,490,659,786,877,938,975,994,1001,333,604,821,990,1117,1208,1269
%N 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.
%C A386215 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.
%F A386215 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
%e A386215 First thirty (1,3)-quartals (2,u,v,w):
%e A386215   m     u      v     w
%e A386215   2     2      9     1
%e A386215   2     3     21     2
%e A386215   2     3     28     1
%e A386215   2     4     39     3
%e A386215   2     4     58     2
%e A386215   2     4     65     1
%e A386215   2     5     63     4
%e A386215   2     5    100     3
%e A386215   2     5    119     2
%e A386215   2     5    126     1
%e A386215   2     6     93     5
%e A386215   2     6    154     4
%e A386215   2     6    191     3
%e A386215   2     6    210     2
%e A386215   2     6    217     1
%e A386215   2     7    129     6
%e A386215   2     7    220     5
%e A386215   2     7    281     4
%e A386215   2     7    318     3
%e A386215   2     7    337     2
%e A386215   2     7    344     1
%e A386215   2     8    171     7
%e A386215   2     8    298     6
%e A386215   2     8    389     5
%e A386215   2     8    450     4
%e A386215   2     8    487     3
%e A386215   2     8    506     2
%e A386215   2     8    513     1
%e A386215   2     9    219     8
%e A386215   2     9    388     7
%e A386215 2^1 + 3^3 = 21^1 + 2^3, so (2,3,21,2) is in the list.
%p A386215 seq(seq(2 + u^3 - w^3, w = u-1 .. 1,-1),u=2..20); # _Robert Israel_, Jul 27 2025
%t A386215 quartals[m_, p_, q_, max_] :=
%t A386215   Module[{ans = {}, lhsD = <||>, lhs, v, u, w, rhs},
%t A386215    For[u = 1, u <= max, u++, lhs = m^p + u^q;
%t A386215     AssociateTo[lhsD, lhs -> Append[Lookup[lhsD, lhs, {}], u]];];
%t A386215    For[v = m + 1, v <= max, v++,
%t A386215     For[w = 1, w <= max, w++, rhs = v^p + w^q;
%t A386215       If[KeyExistsQ[lhsD, rhs],
%t A386215        Do[AppendTo[ans, {m, u, v, w}], {u, lhsD[rhs]}];];];];
%t A386215    ans = SortBy[ans, #[[2]] &];
%t A386215    Do[Print["Solution ", i, ": ", ans[[i]], " (", m, "^", p, " + ",
%t A386215      ans[[i, 2]], "^", q, " = ", ans[[i, 3]], "^", p, " + ",
%t A386215      ans[[i, 4]], "^", q, " = ", m^p + ans[[i, 2]]^q, ")"], {i,
%t A386215      Length[ans]}]; ans];
%t A386215 solns = quartals[2, 1, 3, 2000]  (* solutions restricted to v<2000 *)
%t A386215 Grid[solns]
%t A386215 u1 = Map[#[[2]] &, solns]  (*u, A003057 *)
%t A386215 v1 = Map[#[[3]] &, solns]  (*v, A386215 *)
%t A386215 w1 = Map[#[[4]] &, solns]  (*w, A004736 *)
%t A386215 (* _Peter J. C. Moses_, Jun 20 2025 *)
%Y A386215 Cf. A003057, A004736, A385882.
%K A386215 nonn
%O A386215 1,1
%A A386215 _Clark Kimberling_, Jul 22 2025