A309961 - OEIS

%I #15 Jan 24 2023 10:13:18

%S 6,7,9,12,13,15,17,20,22,26,28,31,33,34,35,42,43,48,49,50,51,53,56,58,

%T 61,62,63,67,68,69,70,71,72,75,78,79,84,85,87,89,90,92,94,96,97,98,

%U 103,104,105,106,107,114,115,117,120,123,130,133,134,136,139,140,141,142,143,151

%N Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 1.

%H Charles R Greathouse IV, <a href="/A309961/b309961.txt">Table of n, a(n) for n = 1..10000</a>

%F A060838(a(n)) = 1.

%o (PARI) for(k=1, 200, if(ellanalyticrank(ellinit([0, 0, 0, 0, -432*k^2]))[1]==1, 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=ellinit([0, 16*r^2]), eri=ellrankinit(E), mwr=ellrank(eri), ar); if(r<6 || mwr[1]==0, return(0)); if(mwr[2]>1, return(0)); ar=ellanalyticrank(E)[1]; if(ar==0, return(0)); for(effort=1, 99, mwr=ellrank(eri, effort); if(mwr[1]>1 || mwr[2]<1, return(0), mwr[1]==mwr[2] && mwr[1]==1, return(1))); error("unknown (",ar==1," on the BSD conjecture)") \\ _Charles R Greathouse IV_, Jan 24 2023

%Y Subsequence of A159843.

%Y Cf. A060748, A060838, A309960 (rank 0), A309962 (rank 2), A309963 (rank 3), A309964 (rank 4).

%K nonn

%O 1,1

%A _Seiichi Manyama_, Aug 25 2019