A002833 -id:A002833 - cp's OEIS frontend

A174627 Partial sums of A002833.

0, 1, 3, 11, 55, 545, 14619, 1363847

Offset: 0

Jonathan Vos Post, Mar 24 2010

Partial sums of threshold functions of n variables. The subsequence of primes in this partial sum begins: 3, 11, 1363847.

			a(7) = 0 + 1 + 2 + 8 + 44 + 490 + 14074 + 1349228 = 1363847 is prime.

Cf. A002833.

a(n) = SUM[i=0..n] A002833(i).

A179127 Numbers n for which the order of Tate-Shafarevich group Ш (Sha) of the elliptic curve y^2=x^3+n is 4.

Original entry on oeis.org

123, 174, 214, 231, 286, 362, 383, 445, 487, 510, 527, 546, 566, 571, 608, 627, 669, 706, 718, 734, 741, 762, 805, 914, 942, 965, 970, 1019, 1042, 1059, 1075, 1131, 1155, 1166, 1189, 1203, 1210, 1230, 1236, 1245, 1287, 1320, 1355, 1392, 1397, 1410, 1411

Offset: 1

Artur Jasinski, Jun 30 2010

nonn

For n<123 the order of the Tate-Shafarevich group Ш of the elliptic curve y^2=x^3+n is 1.

J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Cf. A002151, A002153, A002155, A002833, A031507.

Edited by N. J. A. Sloane, Jul 05 2010

A179126 Positive integers m for which the torsion subgroup of the elliptic curve y^2 = x^3 + m has order 3.

Original entry on oeis.org

4, 9, 16, 25, 36, 49, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704

Offset: 1

Artur Jasinski, Jun 30 2010

Apparently equal to the set of integers (A004709(k))^2, k>=2. [This is incorrect, as shown by the terms 256, 576, 1024, 1600, and 2304. - Jianing Song, Aug 25 2022]
From Jianing Song, Aug 25 2022: (Start)
Numbers which are perfect squares (A000290) but not perfect cubes (A000578). This follows from the complete description of the torsion group of y^2 = x^3 + n, using O to denote the point at infinity (see Exercise 10.19 of Chapter X of Silverman's Arithmetic of elliptic curves):
- If n = t^6 is a sixth power, then the torsion group consists of O, (2*t^2,+-3*t^3), (0,+-t^3), and (-t^2, 0).
- If n = t^2 is not a sixth power, then the torsion group consists of O and (0,+-t).
- If n = t^3 is not a sixth power, then the torsion group consists of O and (-t,0).
- If n is of the form -432*t^6, then the torsion group consists of O and (12*t^2,+-36*t^3).
- In all the other cases, the torsion group is trivial. (End)

Jianing Song, Table of n, a(n) for n = 1..10000
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
Joseph H. Silverman, The Arithmetic of Elliptic Curves

Cf. A179125, A002151, A002153, A002155, A002833, A031507, A228532.

isA179126(n) = my(k=ispower(n)); !(k%2) && (k%6) \\ Jianing Song, Aug 25 2022

is_A179126 = lambda n: EllipticCurve([0,n]).torsion_order() == 3  # D. S. McNeil, Jan 04 2011

A179128 Numbers n for which order of Tate-Shafarevich group Ш of the elliptic curve y^2=x^3+n is 5.

Original entry on oeis.org

8798, 9834

Offset: 1

Artur Jasinski, Jun 30 2010

For n<123 order of Tate-Shafarevich group Ш of the elliptic curve y^2=x^3+n is 1.

For #Ш=4 see A179127.

J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Cf. A002151, A002153, A002155, A002833, A031507.

A179129 Numbers k for which order of Tate-Shafarevich group Ш of the elliptic curve y^2=x^3+k is 9.

Original entry on oeis.org

410, 790, 851, 1294, 1383, 1546, 1635, 1735, 1866, 2139, 2167, 2230, 2363, 2419, 2685, 2743, 2757, 2867, 2958, 3021, 3028, 3119, 3355, 3422, 3490, 3630, 3719, 3903, 3962, 4199, 4365, 4421, 4498, 4722, 4731, 4765, 4927, 4954, 4974, 5011, 5018, 5109

Offset: 1

Artur Jasinski, Jun 30 2010

For k<123 order of Tate-Shafarevich group Ш of the elliptic curve y^2=x^3+k is 1.

For #Ш=4 see A179127. For #Ш=5 see A179128.

J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Cf. A002151, A002153, A002155, A002833, A031507, A179127, A179128, A179130.

cp's OEIS Frontend

A174627 Partial sums of A002833.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A179127 Numbers n for which the order of Tate-Shafarevich group Ш (Sha) of the elliptic curve y^2=x^3+n is 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A179126 Positive integers m for which the torsion subgroup of the elliptic curve y^2 = x^3 + m has order 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Sage

A179128 Numbers n for which order of Tate-Shafarevich group Ш of the elliptic curve y^2=x^3+n is 5.

Views

Author

Keywords

Comments

Links

Crossrefs

A179129 Numbers k for which order of Tate-Shafarevich group Ш of the elliptic curve y^2=x^3+k is 9.

Views

Author

Keywords

Comments

Links

Crossrefs