%I
%S 1,6,19,657,21691,489489,9902523,1144421889,1683200989470,
%T 349043376293530,137006962414679910,13293998056584952174157235
%N a(n) = smallest m such that the elliptic curve x^3+y^3=m has rank n, or 1 if no such m exists.
%C From Nick Rogers (rogers(AT)fas.harvard.edu), Jul 03 2003: (Start)
%C I have verified that the first 5 entries are correct; the first two are basically trivial and the third is due to Selmer. I'm not sure who first discovered entries 4 and 5 and I expect that they had been previously proved to be the smallest values.
%C but I have rechecked that they are minimal for their respective rank using a combination of 3descent, MAGMA and John Cremona's program mwrank.
%C There are new smaller values for ranks 6 and 7, namely k = 9902523 has rank 6 and k = 1144421889 has rank 7. 3descent combined with Ian Connell's package apecs for Maple verifies that these are minimal subject to the Birch and SwinnertonDyer conjecture and the Generalized Riemann Hypothesis for Lfunctions associated to elliptic curves.
%C Finally, there are new entries for ranks 8 and 9: k = 1683200989470 has rank 8 and k = 148975046052222390 has rank 9. It seems somewhat likely that the rank 8 example is minimal. (End)
%C The sequence might be finite, even if it is redefined as smallest m such that x^3+y^3=m has rank >= n.  _Jonathan Sondow_, Oct 27 2013
%H Noam D. Elkies, <a href="https://listserv.nodak.edu/cgibin/wa.exe?A2=NMBRTHRY;c51b26cf.0310">Yet more rank records for x^3+y^3=k</a>, Posting to Number Theory List, Oct 19 2003, for a(9)
%H Noam D. Elkies and Nicholas F. Rogers, <a href="https://listserv.nodak.edu/cgibin/wa.exe?A2=NMBRTHRY;76505e0e.0307">Rank records for x^3+y^3=k, cont'd</a>, Posting to Number Theory List, Jul 18 2003, for a(8) and a(9).
%H Noam D. Elkies and Nicholas F. Rogers, <a href="http://link.springer.com/chapter/10.1007/9783540248477_13">Elliptic curves x^3 + y^3 = k of high rank</a>, Algorithmic Number Theory, 6th International Symposium, ANTSVI, Burlington, VT, USA, June 1318, 2004, Proceedings, Springer, Berlin, Heidelberg, 2004, pp. 184193. [<a href="https://arxiv.org/abs/math/0403116">arXiv:math/0403116</a> [math.NT], 2004.]
%H Troy Kessler, <a href="https://listserv.nodak.edu/cgibin/wa.exe?A2=NMBRTHRY;bc34aa4.0104">3 descent on elliptic curve</a>, Posting to Number Theory List, Apr 22, 2001.
%H Nick Rogers, <a href="http://projecteuclid.org/euclid.em/1045759524">Rank computations for the congruent number elliptic curves</a>. Experimental Mathematics 9 (2000), no. 4, 591594.
%o (PARI) {a(n) = my(k=1); while(ellanalyticrank(ellinit([0, 0, 0, 0, 432*k^2]))[1]<>n, k++); k} \\ _Seiichi Manyama_, Aug 24 2019
%Y Positions of records in A060838.
%Y Cf. A230564.
%K nonn,nice
%O 0,2
%A _N. J. A. Sloane_, Apr 23 2001
%E Definition clarified by _Jonathan Sondow_, Oct 27 2013
%E a(10)a(11) from _Amiram Eldar_ were taken from the paper by Elkies & Rogers, Jul 27 2017.
%E Escape clause added by _N. J. A. Sloane_, Oct 26 2017
