cp's OEIS Frontend

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.

Showing 1-5 of 5 results.

A309960 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 0.

Original entry on oeis.org

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, 46, 47, 52, 54, 55, 57, 59, 60, 64, 66, 73, 74, 76, 77, 80, 81, 82, 83, 88, 93, 95, 99, 100, 101, 102, 108, 109, 111, 112, 113, 116, 118, 119, 121, 122, 125, 128, 129, 131, 135, 137
Offset: 1

Views

Author

Seiichi Manyama, Aug 25 2019

Keywords

Links

Crossrefs

Complement of A159843 \ A000578.
Cf. A060748, A060838, A309961 (rank 1), 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]==0, 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, 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

Formula

A060838(a(n)) = 0.

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

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Aug 25 2019

Keywords

Links

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.

A309962 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 2.

Original entry on oeis.org

19, 30, 37, 65, 86, 91, 110, 124, 126, 127, 132, 152, 153, 163, 182, 183, 201, 203, 209, 210, 217, 218, 219, 240, 246, 254, 271, 273, 282, 296, 309, 335, 342, 345, 348, 370, 379, 390, 397, 399, 407, 420, 433, 435, 436, 446, 453, 462, 468, 469, 477, 497, 498, 506, 513, 520, 523, 554
Offset: 1

Views

Author

Seiichi Manyama, Aug 25 2019

Keywords

Links

Crossrefs

Subsequence of A159843.
Cf. A060748, A060838, A309960 (rank 0), A309961 (rank 1), A309963 (rank 3), A309964 (rank 4).

Programs

  • PARI
    for(k=1, 1e3, if(ellanalyticrank(ellinit([0, 0, 0, 0, -432*k^2]))[1]==2, print1(k", ")))

Formula

A060838(a(n)) = 2.

A309964 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 4.

Original entry on oeis.org

21691, 27937, 33193, 34706, 36667, 39331, 45353, 46299, 53265, 55298, 55335, 59295, 59690, 62628, 63147, 64001, 65683, 73963, 78604, 82290, 87653, 90489, 94681, 96139
Offset: 1

Views

Author

Seiichi Manyama, Aug 25 2019

Keywords

Crossrefs

Subsequence of A159843.
Cf. A060748, A060838, A309960 (rank 0), A309961 (rank 1), A309962 (rank 2), A309963 (rank 3).

Programs

  • PARI
    for(k=1, 5e4, if(ellanalyticrank(ellinit([0, 0, 0, 0, -432*k^2]))[1]==4, print1(k", ")))

Formula

A060838(a(n)) = 4.

Extensions

a(18)-a(24) from Maksym Voznyy, Jan 25 2023

A359687 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 5.

Original entry on oeis.org

489489, 525698, 526535, 763002, 903210, 1423214
Offset: 1

Views

Author

Maksym Voznyy and Charles R Greathouse IV, Jan 25 2023

Keywords

Crossrefs

Subsequence of A159843.
Cf. A060748, A060838, A309960 (rank 0), A309961 (rank 1), A309962 (rank 2), A309963 (rank 3), A309964 (rank 4).

Programs

  • PARI
    is(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<489489, return(0)); if(mwr[1]>5 || mwr[2]<5, return(0)); ar=ellanalyticrank(E)[1]; if(ar<2, return(0)); for(effort=1, 99, mwr=ellrank(eri, effort); if(mwr[1]>5 || mwr[2]<5, return(0), mwr[1]==5 && mwr[2]==5, return(1))); Str("unknown; ",ar==5," under BSD conjecture") \\ Charles R Greathouse IV, Jan 25 2023

Formula

A060838(a(n)) = 5.
Showing 1-5 of 5 results.

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