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 A374926 #29 Aug 02 2024 18:49:55 %S A374926 1,2,5,24,113,337,6310,78560,423515,765617 %N A374926 Least k such that the rank of the elliptic curve y^2 = x^3 - x + k^2 is n, or -1 if no such k exists. %C A374926 This family of curves quickly reaches a moderate value of rank with a relatively low "k" parameter. And is fully analyzed in Tadik's work (see link). Tadik finds 11 terms, a rank lower bound and shows the torsion group is always trivial. The evolution of the rank is shown in detail, finding that a(11) <= 1118245045. %C A374926 I have sequentially checked the first 10 terms, thus proving that they are the least k for each rank. %H A374926 Ezra Brown and Bruce T. Myers, <a href="https://personal.math.vt.edu/brown/doc/dioellip.pdf">Elliptic Curves from Mordell to Diophantus and Back</a>, Amer. Math. Monthly 109 (2002), 639-649. %H A374926 Edward Vincent Eikenberg, <a href="https://api.drum.lib.umd.edu/server/api/core/bitstreams/2da5d47d-4958-4871-8b2f-f40abc3b3c03/content">Rational points on some families of Elliptic Curves</a>, PhD thesis, University of Maryland, 2004. %H A374926 Petra Tadik, <a href="https://ami.uni-eszterhazy.hu/uploads/papers/finalpdf/AMI_40_from145to153.pdf">The rank of certain subfamilies of the elliptic curve y^2 = x^3 -x +t^2</a>, Ann. Math. Inform. 40 (2012), 145-153. %e A374926 The curve C[1] = [-1,1^2] has rank one, with generator [1,-1].The rank of C[2] = [-1,2^2] is 2 because it has two generators:PARI> e=ellinit([-1,2^2] );ellgenerators(e) = [[-1, 2], [0, 2]].If k>1, the curve C[k] always has at least two generators: [0,k], [-1,k], then its minimum rank is two. %Y A374926 Cf. A194687, A309060, A309068, A309069. %K A374926 nonn,hard,more %O A374926 1,2 %A A374926 _Jose Aranda_, Jul 24 2024