A062695 -id:A062695 - cp's OEIS frontend

A259680 Let m = A062695(n); a(n) is value of s in decomposition of m defined in Comments.

1, 1, 1, 137, 6, 29, 1, 1, 97, 5, 73, 1, 1, 1, 1, 1, 1, 17, 6, 1, 53, 1, 5, 41, 6, 2, 1, 1, 1, 101, 257, 7, 17, 1, 1, 7, 2, 337, 689, 7, 1, 1, 761, 37, 793, 1, 1, 1, 181, 61, 1, 21, 5, 1, 151, 1, 1, 1, 7, 1, 1, 1145, 2, 1, 11, 7, 2, 1, 593, 1, 1, 1217, 1, 1, 641

Offset: 1

N. J. A. Sloane, Jul 04 2015

nonn

Let m = A062695(n). Write m*y^2 = x^3 - x as m*square = A*B*(A-B)*(A+B) where A and B are the numerator and denominator of x. Then A, B, A-B, A+B have the form s*a^2, t*b^2, u*c^2, v*d^2 for some decomposition of m as s*t*u*v and some natural numbers a,b,c,d. These eight numbers are given in A259680-A259687.

G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340.
G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340. [Annotated, corrected, scanned copy]

Cf. A003273, A006991, A062695, A259680-A259687.

More terms from Jinyuan Wang, Jan 01 2021

A259687 Let m = A062695(n); a(n) is value of d in decomposition of m defined in Comments.

Original entry on oeis.org

1, 1, 1, 81, 5, 7, 1, 1, 13, 1, 11, 1, 185, 1, 1, 7, 1, 27, 1, 1, 9, 1, 9, 9, 11, 3, 15, 325, 1, 11, 17, 1, 1, 1, 1, 1, 5, 25, 33, 11, 7, 47, 801, 5, 193, 1, 1, 1, 19, 11, 13, 25, 21, 17, 635, 5, 37, 1, 1, 1, 1, 177, 23, 1, 1, 43, 9, 1, 5465, 27, 1, 2721, 1, 17

Offset: 1

N. J. A. Sloane, Jul 04 2015

nonn

Let m = A062695(n). Write m*y^2 = x^3 - x as m*square = A*B*(A-B)*(A+B) where A and B are the numerator and denominator of x. Then A, B, A-B, A+B have the form s*a^2, t*b^2, u*c^2, v*d^2 for some decomposition of m as s*t*u*v and some natural numbers a,b,c,d. These eight numbers are given in A259680-A259687.

G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340.
G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340. [Annotated, corrected, scanned copy]

Cf. A003273, A006991, A062695, A259680-A259687.

More terms from Jinyuan Wang, Jan 01 2021

A259681 Let m = A062695(n); a(n) is value of t in decomposition of m defined in Comments.

Original entry on oeis.org

2, 1, 1, 1, 1, 5, 2, 7, 2, 2, 3, 2, 1, 1, 3, 13, 1, 19, 5, 1, 7, 2, 1, 10, 1, 31, 26, 1, 15, 5, 2, 6, 1, 2, 1, 3, 47, 2, 1, 1, 3, 43, 1, 3, 1, 2, 7, 10, 5, 15, 1, 1, 1, 59, 1, 1, 1, 1, 1, 2, 13, 1, 191, 2, 1, 1, 31, 15, 2, 5, 1, 1, 1, 1, 2, 1, 5, 13, 2, 7, 19

Offset: 1

N. J. A. Sloane, Jul 04 2015

nonn

Let m = A062695(n). Write m*y^2 = x^3 - x as m*square = A*B*(A-B)*(A+B) where A and B are the numerator and denominator of x. Then A, B, A-B, A+B have the form s*a^2, t*b^2, u*c^2, v*d^2 for some decomposition of m as s*t*u*v and some natural numbers a,b,c,d. These eight numbers are given in A259680-A259687.

G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340.
G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340. [Annotated, corrected, scanned copy]

Cf. A003273, A006991, A062695, A259680-A259687.

More terms from Jinyuan Wang, Jan 01 2021

A259682 Let m = A062695(n); a(n) is value of u in decomposition of m defined in Comments.

Original entry on oeis.org

1, 1, 5, 1, 23, 1, 7, 1, 1, 3, 1, 1, 257, 5, 1, 23, 1, 1, 1, 1, 1, 1, 79, 1, 71, 1, 17, 457, 1, 1, 1, 1, 1, 7, 21, 1, 1, 1, 1, 103, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 5, 47, 199, 1, 7, 37, 1081, 13, 3, 17, 3, 1, 3, 1, 7, 167, 19, 1, 1, 239, 1, 1, 1, 1, 1, 1, 1, 103

Offset: 1

N. J. A. Sloane, Jul 04 2015

nonn

Let m = A062695(n). Write m*y^2 = x^3 - x as m*square = A*B*(A-B)*(A+B) where A and B are the numerator and denominator of x. Then A, B, A-B, A+B have the form s*a^2, t*b^2, u*c^2, v*d^2 for some decomposition of m as s*t*u*v and some natural numbers a,b,c,d. These eight numbers are given in A259680-A259687.

G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340.
G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340. [Annotated, corrected, scanned copy]

Cf. A003273, A006991, A062695, A259680-A259687.

More terms from Jinyuan Wang, Jan 01 2021

A259683 Let m = A062695(n); a(n) is value of v in decomposition of m defined in Comments.

Original entry on oeis.org

17, 41, 13, 1, 1, 1, 11, 23, 1, 7, 1, 113, 1, 53, 97, 1, 313, 1, 11, 353, 1, 193, 1, 1, 1, 7, 1, 1, 31, 1, 1, 13, 33, 43, 29, 31, 7, 1, 1, 1, 241, 1, 1, 7, 1, 433, 127, 89, 1, 1, 197, 1, 1, 17, 1, 29, 1, 85, 53, 33, 29, 1, 1, 577, 15, 1, 1, 79, 1, 1, 1201, 1, 1241

Offset: 1

N. J. A. Sloane, Jul 04 2015

nonn

Let m = A062695(n). Write m*y^2 = x^3 - x as m*square = A*B*(A-B)*(A+B) where A and B are the numerator and denominator of x. Then A, B, A-B, A+B have the form s*a^2, t*b^2, u*c^2, v*d^2 for some decomposition of m as s*t*u*v and some natural numbers a,b,c,d. These eight numbers are given in A259680-A259687.

G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340.
G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340. [Annotated, corrected, scanned copy]

Cf. A003273, A006991, A062695, A259680-A259687.

More terms from Jinyuan Wang, Jan 01 2021

A259684 Let m = A062695(n); a(n) is value of a in decomposition of m defined in Comments.

Original entry on oeis.org

3, 5, 3, 5, 2, 1, 3, 4, 1, 1, 1, 9, 153, 7, 7, 6, 13, 5, 1, 17, 1, 11, 4, 1, 4, 4, 11, 253, 4, 1, 1, 1, 1, 5, 5, 2, 8, 1, 1, 4, 103, 39, 29, 2, 5, 19, 8, 7, 1, 1, 163, 4, 8, 63, 44, 23, 35, 7, 2, 5, 4, 5, 13, 17, 1, 12, 5, 8, 193, 22, 25, 65, 29, 481, 1, 85, 1

Offset: 1

N. J. A. Sloane, Jul 04 2015

nonn

Let m = A062695(n). Write m*y^2 = x^3 - x as m*square = A*B*(A-B)*(A+B) where A and B are the numerator and denominator of x. Then A, B, A-B, A+B have the form s*a^2, t*b^2, u*c^2, v*d^2 for some decomposition of m as s*t*u*v and some natural numbers a,b,c,d. These eight numbers are given in A259680-A259687.

G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340.
G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340. [Annotated, corrected, scanned copy]

Cf. A003273, A006991, A062695, A259680-A259687.

More terms from Jinyuan Wang, Jan 01 2021

A259685 Let m = A062695(n); a(n) is value of b in decomposition of m defined in Comments.

Original entry on oeis.org

2, 4, 2, 56, 1, 2, 1, 1, 6, 1, 4, 4, 104, 2, 4, 1, 12, 4, 1, 8, 2, 6, 1, 2, 5, 1, 2, 204, 1, 2, 4, 1, 4, 3, 2, 1, 1, 12, 20, 3, 20, 4, 40, 3, 132, 6, 3, 2, 6, 2, 82, 17, 11, 4, 333, 14, 12, 6, 5, 2, 1, 52, 1, 12, 2, 29, 1, 1, 1972, 7, 24, 1504, 20, 360, 10, 2952

Offset: 1

N. J. A. Sloane, Jul 04 2015

nonn

Let m = A062695(n). Write m*y^2 = x^3 - x as m*square = A*B*(A-B)*(A+B) where A and B are the numerator and denominator of x. Then A, B, A-B, A+B have the form s*a^2, t*b^2, u*c^2, v*d^2 for some decomposition of m as s*t*u*v and some natural numbers a,b,c,d. These eight numbers are given in A259680-A259687.

G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340.
G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340. [Annotated, corrected, scanned copy]

Cf. A003273, A006991, A062695, A259680-A259687.

More terms from Jinyuan Wang, Jan 01 2021

A259686 Let m = A062695(n); a(n) is value of c in decomposition of m defined in Comments.

Original entry on oeis.org

1, 3, 1, 17, 1, 3, 1, 3, 5, 1, 5, 7, 7, 3, 1, 1, 5, 11, 1, 15, 5, 7, 1, 1, 1, 1, 1, 7, 1, 9, 15, 1, 1, 1, 1, 5, 9, 7, 17, 1, 97, 7, 799, 11, 49, 17, 1, 3, 1, 1, 63, 1, 1, 55, 161, 3, 1, 1, 1, 1, 1, 161, 7, 1, 1, 1, 1, 7, 3783, 1, 7, 1697, 21, 319, 21, 911, 3, 1

Offset: 1

N. J. A. Sloane, Jul 04 2015

nonn

Let m = A062695(n). Write m*y^2 = x^3 - x as m*square = A*B*(A-B)*(A+B) where A and B are the numerator and denominator of x. Then A, B, A-B, A+B have the form s*a^2, t*b^2, u*c^2, v*d^2 for some decomposition of m as s*t*u*v and some natural numbers a,b,c,d. These eight numbers are given in A259680-A259687.

G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340.
G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340. [Annotated, corrected, scanned copy]

Cf. A003273, A006991, A062695, A259680-A259687.

More terms from Jinyuan Wang, Jan 01 2021

A194687 Least k such that the rank of the elliptic curve y^2 = x^3 - k^2*x is n, or -1 if no such k exists.

Original entry on oeis.org

1, 5, 34, 1254, 29274, 48272239, 6611719866

Offset: 0

Charles R Greathouse IV, Sep 01 2011

Fermat found a(0), Biling found a(1), and Wiman found a(2)-a(4). Rogers found upper bounds on a(5) and a(6) equal to their true value; Rathbun and an unknown author verified them as a(5) and a(6), respectively.

a(7) <= 797507543735, see Rogers 2004.

G. Billing, "Beiträge zur arithmetischen theorie der ebenen kubischen kurven geschlechteeins", Nova Acta Reg. Soc. Sc. Upsaliensis (4) 11 (1938), Nr. 1. Diss. 165 S.
N. Rogers, "Elliptic curves x^3 + y^2 = k with high rank", PhD Thesis in Mathematics, Harvard University (2004).
A. Wiman, "Über rationale Punkte auf Kurven y^2 = x(x^2-c^2)", Acta Math. 77 (1945), pp. 281-320.

Andrej Dujella, Ali S. Janfada, and Sajad Salami, A search for high rank congruent number elliptic curves, Journal of Integer Sequences, Vol. 12 (2009), Article 09.5.8.
Randall L. Rathbun, Posting to NMBRTHRY, Aug 25 2011
N. F. Rogers, Rank computations for the congruent number elliptic curves, Exper. Math. 9:4 (2000), pp. 591-594.
K. Rubin and A. Silverberg, Ranks of elliptic curves, p.464, Table 2.
Mark Watkins, On elliptic curves and random matrix theory, Journal de Theorie des Nombres de Bordeaux
Author?, LfunctionsAndModularFormsII / CentralValues / Rank4

Cf. A062693, A062695, A003273, A309028, A309029, A319510.

r(n)=ellanalyticrank(ellinit([0,0,0,-n^2,0]))[1]
rec=0;for(n=1,1e4,t=r(n);if(t>rec,rec=t;print("r("n") = "t)))

Escape clause added to definition by N. J. A. Sloane, Jul 01 2024

A273929 Numbers that are congruent to {5, 6, 7} mod 8 and are squarefree.

Original entry on oeis.org

5, 6, 7, 13, 14, 15, 21, 22, 23, 29, 30, 31, 37, 38, 39, 46, 47, 53, 55, 61, 62, 69, 70, 71, 77, 78, 79, 85, 86, 87, 93, 94, 95, 101, 102, 103, 109, 110, 111, 118, 119, 127, 133, 134, 141, 142, 143, 149, 151, 157, 158, 159, 165, 166, 167, 173, 174

Offset: 1

Frank M Jackson, Jun 04 2016

nonn

It has been shown, conditional on the Birch Swinnerton-Dyer conjecture, that this sequence is a subset of the primitive congruent numbers (A006991). The union of this sequence with A062695 gives A006991. Also this sequence is the intersection of A047574 and A005117.

The asymptotic density of this sequence is 3/Pi^2 (A104141). - Amiram Eldar, Mar 09 2021

Keith Conrad, The Congruent Number Problem, The Harvard College Mathematics Review, 2008.

Cf. A005117, A006991, A047574, A062695, A104141.

[n: n in [1..250] | n mod 8 in [5, 6, 7] and IsSquarefree(n)]; // Vincenzo Librandi, Jun 06 2016

Select[Range[1000], MemberQ[{5, 6, 7}, Mod[#, 8]] && SquareFreeQ[#] &]

is(n) = n % 8 > 4 && issquarefree(n) \\ Felix Fröhlich, Jun 04 2016

cp's OEIS Frontend

A259680 Let m = A062695(n); a(n) is value of s in decomposition of m defined in Comments.

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A259687 Let m = A062695(n); a(n) is value of d in decomposition of m defined in Comments.

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A259681 Let m = A062695(n); a(n) is value of t in decomposition of m defined in Comments.

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A259682 Let m = A062695(n); a(n) is value of u in decomposition of m defined in Comments.

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A259683 Let m = A062695(n); a(n) is value of v in decomposition of m defined in Comments.

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A259684 Let m = A062695(n); a(n) is value of a in decomposition of m defined in Comments.

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A259685 Let m = A062695(n); a(n) is value of b in decomposition of m defined in Comments.

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A259686 Let m = A062695(n); a(n) is value of c in decomposition of m defined in Comments.

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A194687 Least k such that the rank of the elliptic curve y^2 = x^3 - k^2*x is n, or -1 if no such k exists.

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A273929 Numbers that are congruent to {5, 6, 7} mod 8 and are squarefree.

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