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%I #20 Jan 24 2023 12:35:23
%S 1,2,3,4,5,8,10,11,14,16,18,21,23,24,25,27,29,32,36,38,39,40,41,44,45,
%T 46,47,52,54,55,57,59,60,64,66,73,74,76,77,80,81,82,83,88,93,95,99,
%U 100,101,102,108,109,111,112,113,116,118,119,121,122,125,128,129,131,135,137
%N Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 0.
%H Charles R Greathouse IV, <a href="/A309960/b309960.txt">Table of n, a(n) for n = 1..10000</a>
%F A060838(a(n)) = 0.
%o (PARI) for(k=1, 200, if(ellanalyticrank(ellinit([0, 0, 0, 0, -432*k^2]))[1]==0, print1(k", ")))
%o (PARI) is(n, f=factor(n))=my(c=prod(i=1, #f~, f[i, 1]^(f[i, 2]\3)), r=n/c^3, E, eri, mwr, ar); if(r<6, return(1)); E=ellinit([0, 16*r^2]); eri=ellrankinit(E); mwr=ellrank(eri); if(mwr[1], return(0)); ar=ellanalyticrank(E)[1]; if(ar<2, return(!ar)); for(effort=1, 99, mwr=ellrank(eri, effort); if(mwr[1]>0, return(0), mwr[2]<1, return(1))); "unknown (0 under BSD conjecture)" \\ _Charles R Greathouse IV_, Jan 24 2023
%Y Complement of A159843 \ A000578.
%Y Cf. A060748, A060838, A309961 (rank 1), A309962 (rank 2), A309963 (rank 3), A309964 (rank 4).
%K nonn
%O 1,2
%A _Seiichi Manyama_, Aug 25 2019