A228380 -id:A228380 - cp's OEIS frontend

A117319 Values of n for which Leech's problem "Find two rational right-angled triangles on the same base whose heights are in the ratio n:1" has a solution.

7, 10, 11, 12, 14, 17, 19, 22, 23, 27, 28, 29, 30, 33, 38, 39, 40, 41, 42, 44, 45, 47, 48, 51, 52, 53, 54, 57, 58, 59, 61, 67, 69, 74, 76, 79, 80, 81, 82, 83, 84, 85, 88, 92, 93, 96, 97, 100, 102, 103, 105, 107, 108, 109, 111, 112, 113, 115, 118, 119, 120, 121, 124, 126

Offset: 1

Hugo Pfoertner, Mar 07 2006

nonn

Integers n such that the elliptic curve y^2 = x^3 + (n^2+1)*x^2 + n^2*x has a positive rank.

			a(1)=7 because n=7 is the smallest factor n for which both b^2+h^2 and b^2+(n*h)^2 are squares. The corresponding values of base b and height h are b=A117320(1)=12 and h=A117321(1)=5. 12^2+5^2=169=13^2 and 12^2+(7*5)^2=1369=37^2 are both squares.

Allan MacLeod, Solutions to Leech's problem

Cf. A117320 (bases), A117321 (heights), A228380.

{ isA117319(n) = ellanalyticrank(ellinit([0, n^2+1, 0, n^2, 0]))[1]; } /* Max Alekseyev, Sep 29 2015 */

A228914 Positive integers N such that 1/N = p/q - q/p + r/s - s/r for some positive integers p,q,r,s.

Original entry on oeis.org

4, 5, 9, 12, 15, 20, 21, 22, 24, 26, 29, 30, 31, 32, 34, 35, 36, 37, 38, 40, 43, 44, 53, 55, 56, 58, 59, 60, 62, 64, 66, 67, 68, 69, 70, 71, 74, 76, 77, 80, 82, 84, 86, 87, 88, 90, 91, 92, 94, 95, 102, 103, 104, 105, 106, 108, 109, 110, 112, 113, 115, 117, 122, 123, 129, 131, 132, 133, 135, 136, 137, 138, 139, 140, 141, 143, 144, 147

Offset: 1

Thomas Bokk, Sep 13 2013

nonn

Positive integer N belongs to this sequence if and only if the elliptic curve y^2 = x^3 + (8*N^2+1)*x^2 + 16*N^4*x has positive rank.

Cf. A206334, A228380

{ isA228914(n) = ellanalyticrank(ellinit([0, 8*n^2+1, 0, 16*n^4, 0]))[1]; } /* Max Alekseyev, Dec 30 2015 */

More terms from Max Alekseyev, Sep 13 2013

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

A117319 Values of n for which Leech's problem "Find two rational right-angled triangles on the same base whose heights are in the ratio n:1" has a solution.

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A228914 Positive integers N such that 1/N = p/q - q/p + r/s - s/r for some positive integers p,q,r,s.

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

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