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 A154291 #3 Mar 30 2012 17:22:54 %S A154291 23,31,139,211,239,419,491,499,563,643,743,751,823,1291,1319,1427, %T A154291 1931,2039,2687,2767,3011,3119,3163,3191,3299,3307,3803,3919,4027, %U A154291 4091,4099,4423,4703,4999,5323,5639,5647,6007,6043,6079,6323,6691,6719,6763,7331 %N A154291 Primes of the form 4x^3 + 27y^2, with x<0. %C A154291 For each prime p, the elliptic curve 27y^2 = 4x^3 + p must be solved to determine whether there is an integer solution with x positive. About 2/3 of all primes can be eliminated because p-4x^3 is never divisible by 27. The remaining primes are congruent to +-5 (mod 18). Hence this sequence is a subsequence of A129806. Half of those primes can be eliminated because even when 27 does divide p-4x^3, the quotient must equal 1 (mod 4) in order to be a square. Hence all these primes must equal 23 or 31 (mod 36). James Buddenhagen used APECS and I used Sage to examine the elliptic curves. The first difficult prime is 1831. All the elliptic curves with p = 23 or 31 (mod 36) appear to have trivial torsion and rank 0 or 2. %C A154291 See the link to the Sage/Python program to see how the problem with 1831 was resolved. The first prime producing an elliptic curve of rank 4 is 19427. %H A154291 T. D. Noe, <a href="http://www.sspectra.com/math/A154291.txt">Sage/Python program</a> %e A154291 743 = 4*(-17977)^3 + 27*927735^2 %Y A154291 Cf. A153635, A153636 %K A154291 nonn %O A154291 1,1 %A A154291 _T. D. Noe_, Jan 06 2009, Jun 18 2009, Jun 21 2009