A102535 -id:A102535 - cp's OEIS frontend

A102778 Values of x arising from representations of -n in A102535.

1, 1, -1, -5, 1, 7, -1, -68, 1, 1, -12, 228, -1, -15, -770, -1, 1, 2460, -5166, -1, -135762627, 63, -266, 5, 1, -175, 27728847, -1, -19535516, -15, -3, -11, 52, -1, -9762340, -77, -4, -1812880, 1, 5, -3, -46129720, 9, -3003037440

Offset: 1

N. J. A. Sloane, Mar 17 2005

sign

Allan J. MacLeod, Knight's Problem

A102779 Values of y arising from representations of -n in A102535.

Original entry on oeis.org

2, 3, 6, 12, 4, 33, 10, 105, 15, 5, 77, 364, 15, 136, 8151, 12, 6, 11859, 23715, 21, 2269245880, 209, 825, 18, 7, 288, 2085664428, 28, 196404156, 348, 68, 36, 700, 36, 264208596, 2331, 133, 2700432, 9, 19, 102, 177989847, 11, 5581682432

Offset: 1

N. J. A. Sloane, Mar 17 2005

nonn

Allan J. MacLeod, Knight's Problem

A102809 Values of z arising from representations of -n in A102535.

Original entry on oeis.org

-6, -12, 10, 105, -20, -154, 15, 4403, -40, -30, 364, -4921, 21, 440, 24339, 33, -42, -101639, 272510, 28, 5003773320, -2736, 19866, -230, -56, 25312, -3712351475, 36, 1171388655, 740, 156, 1100, -4277, 45, 551048889, 4554, 228

Offset: 1

N. J. A. Sloane, Mar 17 2005

sign

Allan J. MacLeod, Knight's Problem

A085514 Integers n representable as the product of the sum of three nonzero integers with the sum of their reciprocals: n=(x+y+z)*(1/x+1/y+1/z).

Original entry on oeis.org

1, 9, 10, 11, 14, 15, 18, 26, 29, 30, 31, 34, 35, 37, 38, 42, 43, 44, 48, 52, 53, 54, 55, 57, 59, 62, 63, 64, 67, 69, 70, 71, 73, 74, 75, 76, 82, 84, 85, 86, 90, 92, 93, 94, 95, 96, 98, 100, 101, 102, 103, 105, 106, 108, 111, 112, 116, 117, 122, 125, 126, 127, 128

Offset: 1

Hugo Pfoertner, Jul 19 2003

nonn

See under A086446 for comments and references.

			a(1)=1 because (1+1-1)*(1/1+1/1-1/1)=1.
a(2)=(1+1+1)*(1/1+1/1+1/1)=9.
a(9)=(2-15+78)*(1/2-1/15+1/78)=29.

Seiichi Manyama, Table of n, a(n) for n = 1..1000
A. Bremner, R. K. Guy and R. Nowakowski, Which integers are representable as the product of the sum of three integers with the sum of their reciprocals?, Math. Comp. 61 (1993) 117-130.
Allan J. MacLeod, Knight's Problem
Allan J. MacLeod, Solutions for 11 <= n <= 999 (copy from MacLeod's website)
Nguyen Xuan Tho, What positive integers n can be presented in the form n = (x + y + z)(1/x + 1/y + 1/z)?, Annales Mathematicae et Informaticae 54 (2021).

Cf. A086446 (representation by positive x, y, z), A102535 (representable negative n)

See A102774, A102775, A102777 for values of x, y, z corresponding to values of n >= 11.

Corrected and extended by David J. Rusin, Jul 30 2003

More terms from the MacLeod web site, Mar 17 2005

A309144 Rank of elliptic curve y^2 = x^3 + (n^2 + 6n -3)x^2 - 16nx.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Jul 14 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Allan J. MacLeod, Knight's Problem

Cf. A102535.

{a(n) = ellanalyticrank(ellinit([0, n^2+6*n-3, 0, -16*n, 0]))[1]}

cp's OEIS Frontend

A102778 Values of x arising from representations of -n in A102535.

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A102779 Values of y arising from representations of -n in A102535.

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A102809 Values of z arising from representations of -n in A102535.

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A085514 Integers n representable as the product of the sum of three nonzero integers with the sum of their reciprocals: n=(x+y+z)*(1/x+1/y+1/z).

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A309144 Rank of elliptic curve y^2 = x^3 + (n^2 + 6n -3)x^2 - 16nx.

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cp's OEIS Frontend

A102778 Values of x arising from representations of -n in A102535.

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A102779 Values of y arising from representations of -n in A102535.

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A102809 Values of z arising from representations of -n in A102535.

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A085514 Integers n representable as the product of the sum of three nonzero integers with the sum of their reciprocals: n=(x+y+z)*(1/x+1/y+1/z).

Views

Author

Keywords

Comments

Examples

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Crossrefs

Extensions

A309144 Rank of elliptic curve y^2 = x^3 + (n^2 + 6*n -3)*x^2 - 16*n*x.

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PARI

A309144 Rank of elliptic curve y^2 = x^3 + (n^2 + 6n -3)x^2 - 16nx.