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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 see Comments.

%I A386219 #6 Aug 11 2025 15:24:09
%S A386219 11,23,30,41,60,67,65,102,121,128,95,156,193,212,219,131,222,283,320,
%T A386219 339,346,173,300,391,452,489,508,515,221,390,517,608,669,706,725,732,
%U A386219 275,492,661,788,879,940,977,996,1003,335,606,823,992,1119,1210,1271,1308
%N 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<v; see Comments.
%C A386219 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=4, p=1, q=3.
%e A386219 First 20 (1,3)-quartals (4,u,v,w):
%e A386219   m   u    v   w
%e A386219   4   2   11   1
%e A386219   4   3   23   2
%e A386219   4   3   30   1
%e A386219   4   4   41   3
%e A386219   4   4   60   2
%e A386219   4   4   67   1
%e A386219   4   5   65   4
%e A386219   4   5  102   3
%e A386219   4   5  121   2
%e A386219   4   5  128   1
%e A386219   4   6   95   5
%e A386219   4   6  156   4
%e A386219   4   6  193   3
%e A386219   4   6  212   2
%e A386219   4   6  219   1
%e A386219   4   7  131   6
%e A386219   4   7  222   5
%e A386219   4   7  283   4
%e A386219   4   7  320   3
%e A386219   4   7  339   2
%e A386219 4^1 + 5^3 = 65^1 + 4^3, so (4,5,65,4) is in the list.
%t A386219 quartals[m_, p_, q_, max_] := Module[{ans = {}, lhsD = <||>, lhs, v, u, w, rhs},
%t A386219    For[u = 1, u <= max, u++, lhs = m^p + u^q;
%t A386219     AssociateTo[lhsD, lhs -> Append[Lookup[lhsD, lhs, {}], u]];];
%t A386219    For[v = m + 1, v <= max, v++,
%t A386219     For[w = 1, w <= max, w++, rhs = v^p + w^q;
%t A386219       If[KeyExistsQ[lhsD, rhs],
%t A386219        Do[AppendTo[ans, {m, u, v, w}], {u, lhsD[rhs]}];];];];
%t A386219    ans = SortBy[ans, #[[2]] &];
%t A386219    Do[Print["Solution ", i, ": ", ans[[i]], " (", m, "^", p, " + ",
%t A386219      ans[[i, 2]], "^", q, " = ", ans[[i, 3]], "^", p, " + ",
%t A386219      ans[[i, 4]], "^", q, " = ", m^p + ans[[i, 2]]^q, ")"], {i,
%t A386219      Length[ans]}]; ans];
%t A386219 solns = quartals[4, 1, 3, 2000]
%t A386219 Grid[solns]
%t A386219 (* _Peter J. C. Moses_, Jun 20 2025 *)
%Y A386219 Cf. A385882, A386215, A386217.
%K A386219 nonn
%O A386219 1,1
%A A386219 _Clark Kimberling_, Jul 28 2025
%E A386219 Data corrected by _Sean A. Irvine_, Aug 11 2025