A117319 -id:A117319 - cp's OEIS frontend

A117320 Bases of right triangles that are solutions to Leech's problem A117319.

12, 24, 21, 45, 15, 528, 1155, 40, 468, 60, 455, 165, 275, 1288, 37961, 6160, 399, 420, 17480, 52080, 84, 46123872, 140, 35, 2275, 68207909, 9673916261759, 92008, 840, 827811, 8040130371, 67599, 91, 1505, 144, 308, 1540, 2405, 84082157037959

Offset: 1

Hugo Pfoertner, Mar 07 2006

nonn

Cf. A117319, A117321 [corresponding heights].

A117321 Heights of right triangles that are solutions to Leech's problem A117319.

Original entry on oeis.org

5, 7, 20, 28, 8, 455, 68, 9, 595, 11, 528, 52, 252, 255, 3120, 111, 40, 29, 561, 4199, 13, 874055, 51, 12, 3828, 654280020, 289469937120, 2145, 41, 30940, 3727393780, 520, 60, 312, 17, 75, 741, 1428, 135429570100440, 49610080, 17020, 19, 97240, 585

Offset: 1

Hugo Pfoertner, Mar 07 2006

nonn

Cf. A117319, A117320 [corresponding bases].

A228380 Positive integers N such that N = (p^2+q^2)(r^2-s^2)/((p^2-q^2)(r^2+s^2)) for some positive integers p,q,r,s.

Original entry on oeis.org

1, 6, 13, 16, 18, 22, 23, 32, 33, 35, 36, 37, 41, 42, 43, 44, 45, 46, 50, 51, 53, 57, 58, 59, 60, 61, 63, 67, 69, 70, 74, 75, 77, 78, 79, 80, 83, 84, 85, 86, 88, 89, 90, 93, 94, 95, 96, 97, 98, 102, 104, 110, 112, 116, 117, 118, 119, 122, 123, 124, 126, 128, 132, 133, 134, 137, 138, 141, 142, 143, 152

Offset: 1

Thomas Bokk, Aug 21 2013

nonn

Integer N>1 belongs to this sequence if and only if the elliptic curve y^2 = x^3 - (N^2+1)*x^2 + N^2*x has positive rank.

Cf. A117319

{ isA228380(n) = ellanalyticrank(ellinit([0,-(n^2+1),0,n^2,0]))[1]; } /* Max Alekseyev, Sep 29 2015 */

More terms from Max Alekseyev, Sep 29 2015

A257642 Positive integers N such that there is a triangle with rational sides having area and perimeter both equal N.

Original entry on oeis.org

21, 24, 26, 27, 28, 30, 31, 33, 35, 36, 37, 39, 42, 43, 45, 47, 50, 51, 52, 55, 56, 58, 60, 61, 62, 63, 64, 66, 67, 71, 74, 75, 76, 77, 79, 81, 83, 85, 86, 88, 90, 91, 93, 94, 95, 96, 98, 99, 100, 102, 103, 105, 106, 107, 108, 109, 110, 112, 113, 115, 116, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 137, 138, 141, 143, 145, 147, 148, 149, 150

Offset: 1

Thomas Bokk, Nov 05 2015

nonn

A positive integer N is in the sequence if and only if there exist positive rational numbers x,y such that x*y>1 and 4*x*y*(x+y)/(x*y-1)=N.

Except for N=27, a positive integer N is in this sequence if and only if N>20 and the elliptic curve w^2 = u^3 + N^2*(u+64)^2 has positive rank.

Cf. A135412, A206334, A228914, A117319, A228380.

cp's OEIS Frontend

A117320 Bases of right triangles that are solutions to Leech's problem A117319.

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A117321 Heights of right triangles that are solutions to Leech's problem A117319.

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A228380 Positive integers N such that N = (p^2+q^2)(r^2-s^2)/((p^2-q^2)(r^2+s^2)) for some positive integers p,q,r,s.

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Comments

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PARI

Extensions

A257642 Positive integers N such that there is a triangle with rational sides having area and perimeter both equal N.

Views

Author

Keywords

Comments

Crossrefs

cp's OEIS Frontend

A117320 Bases of right triangles that are solutions to Leech's problem A117319.

Views

Author

Keywords

Crossrefs

A117321 Heights of right triangles that are solutions to Leech's problem A117319.

Views

Author

Keywords

Crossrefs

A228380 Positive integers N such that N = (p^2+q^2)*(r^2-s^2)/((p^2-q^2)*(r^2+s^2)) for some positive integers p,q,r,s.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Extensions

A257642 Positive integers N such that there is a triangle with rational sides having area and perimeter both equal N.

Views

Author

Keywords

Comments

Crossrefs

A228380 Positive integers N such that N = (p^2+q^2)(r^2-s^2)/((p^2-q^2)(r^2+s^2)) for some positive integers p,q,r,s.