This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A103268 #26 Jun 24 2023 01:30:00
%S A103268 3,6,8,96,192,256,624,686,729
%N A103268 Positive integers x such that there exist positive integers y and z satisfying x^3 + y^3 = z^5.
%C A103268 Warning! These terms have not been proved to be correct. There may be missing terms. - _N. J. A. Sloane_, Jan 05 2007
%C A103268 There are no solutions with (x,y,z) relatively prime. [Bruin]
%C A103268 Trivially, if m^3 + n^3 = z^2, then (z*m)^3 + (z*n)^3 = z^5. So from A103254 we can find many solutions. - _James Mc Laughlin_, Jan 30 2007
%C A103268 For max(x,y) < 1.1*10^12, there are no more terms < 1458. Most likely this is true for all x,y. - _Chai Wah Wu_, Jan 15 2016
%H A103268 F. Beukers, <a href="http://dx.doi.org/10.1215/S0012-7094-98-09105-0">The Diophantine equation Ax^p+By^q=Cz^r</a>, Duke Math. J. 91 (1998), 61-88.
%H A103268 Nils Bruin, <a href="http://dx.doi.org/10.1007/10722028_9">On powers as sums of two cubes</a>, in Algorithmic number theory (Leiden, 2000), 169-184, Lecture Notes in Comput. Sci., 1838, Springer, Berlin, 2000.
%e A103268 1 + 2^3 = 3^2 so 3^3 + 6^3 = 3^5 and 3 and 6 are terms.
%e A103268 With max(x,y) < 10^4, we have these [x,y,z] triples: [3, 6, 3], [8, 8, 4], [96, 192, 24], [256, 256, 32], [729, 1458, 81], [1944, 1944, 108], [686, 2058, 98], [3696, 4368, 168], [3072, 6144, 192], [8192, 8192, 256], [2508, 8436, 228], ... - _David Broadhurst_, Jan 30 2007
%e A103268 These are variously immediate consequences of 1 + 1 = 2, 1 + 2^3 = 3^2, 1 + 3^3 = 2^2*7 and, much more unexpectedly, 11^3 + 37^3 = 2^4*3^2*19^2. The last example shows that solutions with a common factor are not completely trivial. [Comment based on email from Alf van der Poorten, Feb 15 2007]
%e A103268 624^3 + 14352^3 = 312^5. - _Chai Wah Wu_, Jan 11 2016
%t A103268 r[z_] := Reduce[x > 0 && y > 0 && x^3 + y^3 == z^5, {x, y}, Integers];
%t A103268 sols = Reap[Do[rz = r[z]; If[rz =!= False, xyz = {x, y, z} /. {ToRules[rz]}; Print[xyz]; Sow[xyz]], {z, 1, 1000}]][[2, 1]] // Flatten[#, 1]&;
%t A103268 sols[[All, 1]] // Union (* _Jean-François Alcover_, Oct 18 2019 *)
%Y A103268 See A114737 for another version.
%K A103268 more,nonn
%O A103268 1,1
%A A103268 _Cino Hilliard_, Mar 20 2005
%E A103268 Corrected by _David Broadhurst_ and others, Jan 30 2007
%E A103268 Term 624 added by _Chai Wah Wu_, Jan 11 2016