A309961 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 1.

6, 7, 9, 12, 13, 15, 17, 20, 22, 26, 28, 31, 33, 34, 35, 42, 43, 48, 49, 50, 51, 53, 56, 58, 61, 62, 63, 67, 68, 69, 70, 71, 72, 75, 78, 79, 84, 85, 87, 89, 90, 92, 94, 96, 97, 98, 103, 104, 105, 106, 107, 114, 115, 117, 120, 123, 130, 133, 134, 136, 139, 140, 141, 142, 143, 151

Offset: 1

Author

Seiichi Manyama, Aug 25 2019

Keywords

• nonn

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A159843.

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

Programs

PARI

for(k=1, 200, if(ellanalyticrank(ellinit([0, 0, 0, 0, -432*k^2]))[1]==1, print1(k", ")))

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

Formula

A060838(a(n)) = 1.