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%I #40 Aug 24 2019 21:27:32
%S 1,2,11,174,2351,28279,975379
%N Smallest k>0 such that the elliptic curve y^2 = x^3 - k has rank n, if k exists.
%C The sequence might be finite, even if it is redefined as smallest k>0 such that the elliptic curve y^2 = x^3 - k has rank >= n. - _Jonathan Sondow_, Sep 26 2013
%C For bounds on later terms see the Gebel link. - _N. J. A. Sloane_, Jul 05 2010
%C See A031507 for the smallest k>0 such that the elliptic curve y^2 = x^3 + k has rank n. - _Jonathan Sondow_, Sep 06 2013
%C See A060951 for the rank of y^2 = x^3 - n. - _Jonathan Sondow_, Sep 10 2013
%C Gebel, Pethö, & Zimmer: "One experimental observation derived from the tables is that the rank r of Mordell's curves grows according to r = O(log |k|/|log log |k||^(2/3))." Hence this fit suggests a(n) >> exp(n (log n)^(1/3)) where >> is the Vinogradov symbol. - _Charles R Greathouse IV_, Sep 10 2013
%H J. Gebel, <a href="/A001014/a001014.txt">Integer points on Mordell curves</a> [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
%H J. Gebel, A. Pethö, H. G. Zimmer, <a href="https://doi.org/10.1023/A:1000281602647">On Mordell's equation</a>, Compositio Mathematica 110 (1998), 335-367. <a href="http://www.ams.org/mathscinet-getitem?mr=1602064">MR1602064</a>.
%H Tom Womack, <a href="http://www.tom.womack.net/maths/mordellc.htm">Minimal-known positive and negative k for Mordell curves of given rank</a>.
%o (PARI) {a(n) = my(k=1); while(ellanalyticrank(ellinit([0, 0, 0, 0, -k]))[1]<>n, k++); k} \\ _Seiichi Manyama_, Aug 24 2019
%Y Cf. A002150, A002152, A002154, A031507, A060951, A179136, A179137.
%K nonn,nice,hard,more
%O 0,2
%A _Noam D. Elkies_
%E Definition clarified by _Jonathan Sondow_, Oct 26 2013
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