A003273 -id:A003273 - cp's OEIS frontend

A248406 Noncongruent squarefree numbers n with A248395(n)/d(n) = -2, where d(n) = A000005(n).

146, 178, 274, 322, 466, 938, 994, 1002, 1234, 1394, 1498, 1714, 1866, 1906, 2066, 2098, 2162, 2194, 2386, 2578, 2586, 2786, 2794, 2962, 3002, 3034, 3218, 3266, 3346, 3658, 3682, 3914, 3986, 4090, 4130, 4594, 4738, 4786, 4946, 5170, 5210, 5234, 5266, 5402, 5458

Offset: 1

N. J. A. Sloane, Oct 20 2014

nonn

J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

Cf. A000005, A003273, A248394, A248395, A248397-A248406.

More terms from Amiram Eldar, Oct 13 2019

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

Original entry on oeis.org

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

A072068 Number of integer solutions to the equation 2x^2+y^2+8z^2=m for an odd number m=2n-1.

Original entry on oeis.org

2, 4, 0, 0, 10, 12, 0, 0, 16, 12, 0, 0, 10, 16, 0, 0, 16, 24, 0, 0, 32, 12, 0, 0, 18, 24, 0, 0, 16, 36, 0, 0, 32, 12, 0, 0, 16, 28, 0, 0, 34, 36, 0, 0, 48, 24, 0, 0, 16, 36, 0, 0, 32, 36, 0, 0, 32, 24, 0, 0, 26, 24, 0, 0

Offset: 1

T. D. Noe, Jun 13 2002

nonn

Related to primitive congruent numbers A006991.
Assuming the Birch and Swinnerton-Dyer conjecture, the odd number 2n-1 is a congruent number if it is squarefree and a(n) = 2*A072069(n).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			a(2) = 4 because (1,1,0), (-1,1,0), (1,-1,0) and (-1,-1,0) are solutions when m=3.
G.f. = 2*x + 4*x^2 + 10*x^5 + 12*x^6 + 16*x^9 + 12*x^10 + 10*x^13 + 16*x^14 + 16*x^17 + ...
G.f. = 2*q + 4*q^3 + 10*q^9 + 12*q^11 + 16*q^17 + 12*q^19 + 10*q^25 + 16*q^27 + ...

T. D. Noe, Table of n, a(n) for n=1..10000
Clay Mathematics Institute, The Birch and Swinnerton-Dyer Conjecture
Department of Pure Maths., Univ. Sheffield, Pythagorean triples and the congruent number problem
Karl Rubin, Elliptic curves and right triangles
J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

Cf. A006991, A003273, A072069, A072070, A072071, A080917.

maxN=128; soln1=Table[0, {maxN/2}]; xMax=Ceiling[Sqrt[maxN/2]]; yMax=Ceiling[Sqrt[maxN]]; zMax=Ceiling[Sqrt[maxN/8]]; Do[n=2x^2+y^2+8z^2; If[OddQ[n]&&nA072068 = CoefficientList[s, x] // Rest (* Jean-François Alcover, Feb 16 2015, after Michael Somos *)

{a(n) = my(A); n--; if( n<0, 0, A = x * O(x^n); polcoeff( 2 * eta(x^2 + A)^5 * eta(x^8 + A)^7 / (eta(x + A)^2 * eta(x^4 + A)^5 * eta(x^16 + A)^2), n))}; /* Michael Somos, Dec 26 2019 */

Expansion of 2 * x * phi(x) * psi(x^4) * phi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Jun 08 2012

Expansion of 2 * q^(1/2) * eta(q^2)^5 * eta(q^8)^7 / (eta(q)^2 * eta(q^4)^5 * eta(q^16)^2) in powers of q. - Michael Somos, Feb 19 2015

A072069 Number of integer solutions to the equation 2x^2+y^2+32z^2=m for an odd number m=2n-1.

Original entry on oeis.org

2, 4, 0, 0, 6, 4, 0, 0, 4, 4, 0, 0, 2, 8, 0, 0, 12, 8, 0, 0, 16, 12, 0, 0, 10, 16, 0, 0, 12, 20, 0, 0, 16, 4, 0, 0, 12, 12, 0, 0, 14, 20, 0, 0, 20, 8, 0, 0, 4, 20, 0, 0, 8, 12, 0, 0, 24, 8, 0, 0, 14, 8, 0, 0

Offset: 1

T. D. Noe, Jun 13 2002

nonn

Related to primitive congruent numbers A006991.
Assuming the Birch and Swinnerton-Dyer conjecture, the odd number 2n-1 is a congruent number if it is squarefree and 2 a(n) = A072068(n).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			a(2) = 4 because (1,1,0), (-1,1,0), (1,-1,0) and (-1,-1,0) are solutions when m=3.
G.f. = 2*x + 4*x^2 + 6*x^5 + 4*x^6 + 4*x^9 + 4*x^10 + 2*x^13 + 8*x^14 + ... - _Michael Somos_, Dec 26 2019
G.f. = 2*q + 4*q^3 + 6*q^9 + 4*q^11 + 4*q^17 + 4*q^19 + 2*q^25 + 8*q^27 + 12*q^33
+ ...

J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

T. D. Noe, Table of n, a(n) for n=1..10000
Clay Mathematics Institute, The Birch and Swinnerton-Dyer Conjecture
Department of Pure Maths., Univ. Sheffield, Pythagorean triples and the congruent number problem
Karl Rubin, Elliptic curves and right triangles

Cf. A006991, A003273, A072068, A072070, A072071, A080918.

maxN=128; soln2=Table[0, {maxN/2}]; xMax=Ceiling[Sqrt[maxN/2]]; yMax=Ceiling[Sqrt[maxN]]; zMax=Ceiling[Sqrt[maxN/32]]; Do[n=2x^2+y^2+32z^2; If[OddQ[n]&&n

{a(n) = my(A); n--; if( n<0, 0, A = x * O(x^n); polcoeff( 2 * eta(x^2 + A)^5 * eta(x^8 + A)^2 * eta(x^32 + A)^5 / (eta(x + A)^2 * eta(x^4 + A)^3 * eta(x^16 + A)^2 * eta(x^64 + A)^2), n))}; /* Michael Somos, Dec 26 2019 */

Expansion of 2 * x * phi(x) * psi(x^4) * phi(x^16) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Jun 08 2012

Expansion of 2 * q^(1/2) * eta(q^2)^5 * eta(q^8)^2 * eta(q^32)^5 / (eta(q)^2 * eta(q^4)^3 * eta(q^16)^2 * eta(q^64)^2) in powers of q. - Michael Somos, Dec 26 2019

A072070 Number of integer solutions to the equation 4x^2 + y^2 + 8z^2 = n.

Original entry on oeis.org

1, 2, 0, 0, 4, 4, 0, 0, 6, 6, 0, 0, 8, 12, 0, 0, 12, 8, 0, 0, 8, 8, 0, 0, 8, 14, 0, 0, 16, 4, 0, 0, 6, 16, 0, 0, 12, 20, 0, 0, 24, 8, 0, 0, 8, 20, 0, 0, 24, 18, 0, 0, 24, 12, 0, 0, 0, 16, 0, 0, 16, 20, 0, 0, 12, 8, 0, 0, 16, 16, 0, 0, 30, 32, 0, 0, 24, 16, 0, 0, 24, 18, 0, 0, 16, 24, 0, 0, 24, 16

Offset: 0

T. D. Noe, Jun 13 2002

nonn

Related to primitive congruent numbers A006991.
Assuming the Birch and Swinnerton-Dyer conjecture, the even number 2n is a congruent number if it is squarefree and a(n) = 2 A072071(n).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			a(4) = 4 because (1, 0, 0), (-1, 0, 0), (0, 2, 0) and (0, -2, 0) are solutions.
G.f. = 1 + 2*q + 4*q^4 + 4*q^5 + 6*q^8 + 6*q^9 + 8*q^12 + 12*q^13 + 12*q^16 + 8*q^17 + ...

J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

T. D. Noe, Table of n, a(n) for n = 0..10000
Clay Mathematics Institute, The Birch and Swinnerton-Dyer Conjecture
Department of Pure Maths., Univ. Sheffield, Pythagorean triples and the congruent number problem
Karl Rubin, Elliptic curves and right triangles
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A006991, A003273, A072068, A072069, A072071, A080917.

maxN=128; soln3=Table[0, {maxN/2}]; xMax=Ceiling[Sqrt[maxN/8]]; yMax=Ceiling[Sqrt[maxN/2]]; zMax=Ceiling[Sqrt[maxN/16]]; Do[n=4x^2+y^2+8z^2; If[n>0&&n<=maxN/2, s=8; If[x==0, s=s/2]; If[y==0, s=s/2]; If[z==0, s=s/2]; soln3[[n]]+=s], {x, 0, xMax}, {y, 0, yMax}, {z, 0, zMax}]
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^4] EllipticTheta[ 3, 0, q^8], {q, 0, n}]; (* Michael Somos, Jul 23 2018 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^-2 * eta(x^2 + A)^5 * eta(x^4 + A)^-4 * eta(x^8 + A)^3 * eta(x^16 + A)^3 * eta(x^32 + A)^-2, n))}; /* Michael Somos, Feb 11 2003 */

Expansion of phi(q) * phi(q^4) * phi(q^8) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Jun 09 2012
Euler transform of period 32 sequence [2, -3, 2, 1, 2, -3, 2, -2, 2, -3, 2, 1, 2, -3, 2, -5, 2, -3, 2, 1, 2, -3, 2, -2, 2, -3, 2, 1, 2, -3, 2, -3, ...]. - Michael Somos, Feb 11 2003
a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A014455(n). - Michael Somos, Jun 08 2012
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(7/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A080917. - Michael Somos, Jul 23 2018

A072071 Number of integer solutions to the equation 4x^2+y^2+32z^2=n.

Original entry on oeis.org

1, 2, 0, 0, 4, 4, 0, 0, 4, 2, 0, 0, 0, 4, 0, 0, 4, 4, 0, 0, 8, 0, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 6, 4, 0, 0, 12, 12, 0, 0, 16, 8, 0, 0, 0, 12, 0, 0, 8, 10, 0, 0, 24, 4, 0, 0, 0, 12, 0, 0, 0, 12, 0, 0, 12, 8, 0, 0, 16, 8, 0, 0, 20, 12, 0, 0, 0, 8, 0, 0, 8, 6, 0, 0, 16, 16, 0, 0, 0, 4, 0, 0, 0, 8, 0, 0, 8

Offset: 0

T. D. Noe, Jun 13 2002

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Related to primitive congruent numbers A006991.
Assuming the Birch and Swinnerton-Dyer conjecture, the even number 2n is a congruent number if it is squarefree and 2 a(n) = A072070(n).

			a(4) = 4 because (1,0,0), (-1,0,0), (0,2,0) and (0,-2,0) are solutions.
1 + 2*x + 4*x^4 + 4*x^5 + 4*x^8 + 2*x^9 + 4*x^13 + 4*x^16 + 4*x^17 + 8*x^20 + ...

J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

T. D. Noe, Table of n, a(n) for n = 0..10000
Clay Mathematics Institute, The Birch and Swinnerton-Dyer Conjecture
Department of Pure Maths., Univ. Sheffield, Pythagorean triples and the congruent number problem
Karl Rubin, Elliptic curves and right triangles
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A006991, A003273, A072068, A072069, A072070.

J12[q_] := Sum[q^n^2, {n, -10, 10}]; CoefficientList[Series[J12[q]J12[q^4]J12[q^32], {q, 0, 100}], q]

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^-2 * eta(x^2 + A)^5 * eta(x^4 + A)^-4 * eta(x^8 + A)^5 * eta(x^16 + A)^-2 * eta(x^32 + A)^-2 * eta(x^64 + A)^5 * eta(x^128 + A)^-2, n))}

Expansion of phi(x) * phi(x^4) * phi(x^32) in powers of x where phi() is a Ramanujan theta function.

a(4*n + 2) = a(4*n + 3) = 0. - Michael Somos, Jun 08 2012

More terms from Vladeta Jovovic, Jun 16 2002

A165564 Numbers which are not congruent numbers, i.e., positive integers which are not the area of any right triangle with rational sides.

Original entry on oeis.org

1, 2, 3, 4, 8, 9, 10, 11, 12, 16, 17, 18, 19, 25, 26, 27, 32, 33, 35, 36, 40, 42, 43, 44, 48, 49, 50, 51, 57, 58, 59, 64, 66, 67, 68, 72, 73, 74, 75, 76, 81, 82, 83, 89, 90, 91, 97, 98, 99, 100, 104, 105, 106, 107, 108, 113, 114, 115, 121, 122, 123, 128, 129, 130

Offset: 1

Jose Brox (brox(AT)agt.cie.uma.es), Sep 22 2009

nonn

It is known that every positive integer is the area of some triangle with rational sides. See the survey by Top and Yui. - Jonathan Sondow, Nov 15 2017

Alter, Ronald; Curtz, Thaddeus B.; Kubota, K. K. Remarks and results on congruent numbers. Proceedings of the Third Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1972), pp. 27-35. Florida Atlantic Univ., Boca Raton, Fla., 1972. MR0349554 (50 #2047). - From N. J. A. Sloane, Apr 28 2012

Michel Marcus, Table of n, a(n) for n = 1..4258
Boris Iskra, Non-congruent numbers with arbitrarily many prime factors congruent to 3 modulo 8, Proc. Japan Acad. Ser. A Math. Sci., Volume 72, Number 7 (1996), 168-169.
J. Top and N. Yui, Congruent number problems and their variants, Mathematical Sciences Research Institute Publications, 44 (2008), 613-639.

Complement of A003273.

Integers \ { A003273 }.

Name corrected by Jonathan Sondow, Nov 15 2017

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

A319510 Rank of elliptic curve y^2 = x^3 - n^2 * x.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 2, 0, 0, 1, 1, 1, 0, 2, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 2, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0

Offset: 1

Seiichi Manyama, Sep 24 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A003273, A060952, A062693, A062695, A194687, A273929.

{a(n) = ellanalyticrank(ellinit([0, 0, 0, -n^2, 0]))[1]}

a(n) = A060952(n^2).
a(A003273(n)) > 0.
a(A194687(n)) = n.
Empirical: a(n) = a(4*n). - Jose Aranda, Jul 02 2024

cp's OEIS Frontend

A248406 Noncongruent squarefree numbers n with A248395(n)/d(n) = -2, where d(n) = A000005(n).

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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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A072068 Number of integer solutions to the equation 2x^2+y^2+8z^2=m for an odd number m=2n-1.

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A072069 Number of integer solutions to the equation 2x^2+y^2+32z^2=m for an odd number m=2n-1.

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A072070 Number of integer solutions to the equation 4*x^2 + y^2 + 8*z^2 = n.

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Programs

Mathematica

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A072071 Number of integer solutions to the equation 4x^2+y^2+32z^2=n.

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Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A165564 Numbers which are not congruent numbers, i.e., positive integers which are not the area of any right triangle with rational sides.

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A072070 Number of integer solutions to the equation 4x^2 + y^2 + 8z^2 = n.