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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 see Comments.

%I A386217 #10 Aug 11 2025 15:22:23
%S A386217 10,22,29,40,59,66,64,101,120,127,94,155,192,211,218,130,221,282,319,
%T A386217 338,345,172,299,390,451,488,507,514,220,389,516,607,668,705,724,731,
%U A386217 274,491,660,787,878,939,976,995,1002,334,605,822,991,1118,1209,1270,1307
%N 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<v; see Comments.
%C A386217 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=3, p=1, q=3.
%F A386217 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
%F A386217 a(n) = A385882(n)+2 = A386215(n)+1 = A386219(n)-1. - _Pontus von Brömssen_, Aug 04 2025
%e A386217 First 20 (1,3)-quartals (3,u,v,w):
%e A386217    m   u    v   w
%e A386217    3   2   10   1
%e A386217    3   3   22   2
%e A386217    3   3   29   1
%e A386217    3   4   40   3
%e A386217    3   4   59   2
%e A386217    3   4   66   1
%e A386217    3   5   64   4
%e A386217    3   5  101   3
%e A386217    3   5  120   2
%e A386217    3   5  127   1
%e A386217    3   6   94   5
%e A386217    3   6  155   4
%e A386217    3   6  192   3
%e A386217    3   6  211   2
%e A386217    3   6  218   1
%e A386217    3   7  130   6
%e A386217    3   7  221   5
%e A386217    3   7  282   4
%e A386217    3   7  319   3
%e A386217    3   7  338   2
%e A386217 3^1 + 4^3 = 40^1 + 3^3, so (3,4,40,3) is in the list.
%t A386217 quartals[m_, p_, q_, max_] := Module[{ans = {}, lhsD = <||>, lhs, v, u, w, rhs},
%t A386217    For[u = 1, u <= max, u++, lhs = m^p + u^q;
%t A386217     AssociateTo[lhsD, lhs -> Append[Lookup[lhsD, lhs, {}], u]];];
%t A386217    For[v = m + 1, v <= max, v++,
%t A386217     For[w = 1, w <= max, w++, rhs = v^p + w^q;
%t A386217       If[KeyExistsQ[lhsD, rhs],
%t A386217        Do[AppendTo[ans, {m, u, v, w}], {u, lhsD[rhs]}];];];];
%t A386217    ans = SortBy[ans, #[[2]] &];
%t A386217    Do[Print["Solution ", i, ": ", ans[[i]], " (", m, "^", p, " + ",
%t A386217      ans[[i, 2]], "^", q, " = ", ans[[i, 3]], "^", p, " + ",
%t A386217      ans[[i, 4]], "^", q, " = ", m^p + ans[[i, 2]]^q, ")"], {i,
%t A386217      Length[ans]}]; ans];
%t A386217 solns = quartals[3, 1, 3, 2000]
%t A386217 Grid[solns]
%t A386217 (* _Peter J. C. Moses_, Jun 21 2025 *)
%Y A386217 Cf. A385882, A386215, A386219.
%K A386217 nonn
%O A386217 1,1
%A A386217 _Clark Kimberling_, Jul 28 2025
%E A386217 Data corrected by _Sean A. Irvine_, Aug 01 2025