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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 sorted by u, then v.

%I A385882 #6 Jul 27 2025 16:27:15
%S A385882 8,20,27,38,57,64,62,99,118,125,92,153,190,209,216,128,219,280,317,
%T A385882 336,343,170,297,388,449,486,505,512,218,387,514,605,666,703,722,729,
%U A385882 272,489,658,785,876,937,974,993,1000,332,603,820,989,1116,1207,1268,1305
%N 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<v, sorted by u, then v.
%C A385882 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 = 1, p = 1, q = 3.
%e A385882 First thirty (1,3)-quartals (1,u,v,w):
%e A385882   m   u    v   w
%e A385882   1   2    8   1
%e A385882   1   3   20   2
%e A385882   1   3   27   1
%e A385882   1   4   38   3
%e A385882   1   4   57   2
%e A385882   1   4   64   1
%e A385882   1   5   62   4
%e A385882   1   5   99   3
%e A385882   1   5  118   2
%e A385882   1   5  125   1
%e A385882   1   6   92   5
%e A385882   1   6  153   4
%e A385882   1   6  190   3
%e A385882   1   6  209   2
%e A385882   1   6  216   1
%e A385882   1   7  128   6
%e A385882   1   7  219   5
%e A385882   1   7  280   4
%e A385882   1   7  317   3
%e A385882   1   7  336   2
%e A385882   1   7  343   1
%e A385882   1   8  170   7
%e A385882   1   8  297   6
%e A385882   1   8  388   5
%e A385882   1   8  449   4
%e A385882   1   8  486   3
%e A385882   1   8  505   2
%e A385882   1   8  512   1
%e A385882   1   9  218   8
%e A385882   1   9  387   7
%e A385882 1^1 + 4^3 = 57^1 + 2^3, so (1,4,57,2) is in the list.
%t A385882 quartals[m_, p_, q_, max_] := Module[{ans = {}, lhsD = <||>, lhs, v, u, w, rhs},
%t A385882    For[u = 1, u <= max, u++, lhs = m^p + u^q;
%t A385882     AssociateTo[lhsD, lhs -> Append[Lookup[lhsD, lhs, {}], u]];];
%t A385882    For[v = m + 1, v <= max, v++,
%t A385882     For[w = 1, w <= max, w++, rhs = v^p + w^q; If[KeyExistsQ[lhsD, rhs],
%t A385882        Do[AppendTo[ans, {m, u, v, w}], {u, lhsD[rhs]}];];];];
%t A385882    ans = SortBy[ans, #[[2]] &];
%t A385882    Do[Print["Solution ", i, ": ", ans[[i]], " (", m, "^", p, " + ",
%t A385882      ans[[i, 2]], "^", q, " = ", ans[[i, 3]], "^", p, " + ",
%t A385882      ans[[i, 4]], "^", q, " = ", m^p + ans[[i, 2]]^q, ")"], {i,
%t A385882      Length[ans]}]; ans];
%t A385882 solns = quartals[1, 1, 3, 2000] (* Solutions restricted to v<2000 *)
%t A385882 Grid[solns]
%t A385882 u1 = Map[#[[2]] &, solns]   (*u, A003057 *)
%t A385882 v1 = Map[#[[3]] &, solns]   (*v, A385882 *)
%t A385882 w1 = Map[#[[4]] &, solns]   (*w, A004736 *)
%t A385882 (* _Peter J. C. Moses_, Jun 20 2025 *)
%Y A385882 Guide to related sequences:
%Y A385882    m |    u    |    v    |    w
%Y A385882    --+---------+---------+--------
%Y A385882    1 | A003057 | A385882 | A004736
%Y A385882    2 | A003057 | A386215 | A004736
%Y A385882    3 | A003057 | A386217 | A004736
%Y A385882    4 | A003057 | A386219 | A004736
%Y A385882   --+---------+---------+---------
%Y A385882 Cf. A003057, A004736.
%K A385882 nonn
%O A385882 1,1
%A A385882 _Clark Kimberling_, Jul 21 2025