A085514 -id:A085514 - cp's OEIS frontend

A102774 Values of x arising from representations of n >= 11 in A085514.

1, 3, 1, 2, 1, 5, 10, 3, 1, 3, 7, 90, 561, 24, 1155, 70633, 5, 15, 55, 1, 14, 2, 5075, 7, 1, 5, 4823, 110754153315, 4, 1, 5, 10153, 14, 119, 2, 5, 3, 1386, 231, 106865, 571064, 102, 6816, 1221, 8, 39, 385, 14156395253, 3, 3, 2392, 15351, 2241050

Offset: 1

N. J. A. Sloane, Mar 17 2005

nonn

Allan J. MacLeod, Knight's Problem

A102775 Values of y arising from representations of n >= 11 in A085514.

Original entry on oeis.org

2, 10, 2, 3, 6, -26, 77, -11, 6, 10, -22, 391, 6450, -65, -2109, -3329130, -464, -1190, 595, 14, -174, 15, 128050, 10, -40, -68, -7458, -34644001047040, -33, -20, 22, -58656, -99, 475, -78, -138, -336, 35226, -725, -144690, -1799160

Offset: 1

N. J. A. Sloane, Mar 17 2005

sign

Allan J. MacLeod, Knight's Problem

A102777 Values of z arising from representations of n >= 11 in A085514.

Original entry on oeis.org

3, 15, 6, 15, 14, 195, 165, 132, 21, 65, 385, 2210, 13889, 1640, 110770, 6685382, 720, 1974, 2002, 35, 1015, 85, 160602, 238, 104, 420, 930930, 42290521588224, 348, 95, 270, 957719, 1309, 7106, 247, 570, 592, 81473, 31350, 37646466

Offset: 1

N. J. A. Sloane, Mar 17 2005

nonn

Allan J. MacLeod, Knight's Problem

A102535 Integers n such that -n is 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

4, 10, 11, 12, 18, 19, 20, 22, 25, 28, 29, 30, 31, 32, 36, 39, 40, 42, 43, 44, 48, 50, 51, 52, 54, 56, 58, 59, 61, 67, 69, 70, 72, 76, 78, 84, 85, 86, 88, 89, 91, 92, 95, 96, 100, 101, 102, 103, 104, 105, 107, 108, 109, 112, 113, 115, 116, 120, 122, 123

Offset: 1

N. J. A. Sloane, Mar 17 2005

nonn

Also numbers k such that A309144(k) > 0. - Seiichi Manyama, Jul 14 2019

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 1 <= n <= 1000 (copy from MacLeod's website)

Cf. A085514, A102778, A102779, A102809, A309144.

A309142 Rank of elliptic curve y^2 = x^3 + (n^2 - 6n -3)x^2 + 16nx.

Original entry on oeis.org

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

Offset: 10

Seiichi Manyama, Jul 14 2019

nonn

a(n) is undefined for n = 0, 1 or 9.

Jinyuan Wang, Table of n, a(n) for n = 10..5000
Andrew Bremner, Richard K. Guy, Richard J. 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

Cf. A085514, A086446, A309143, A309144, A309146.

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

A086446 Integers representable as the product of the sum of three positive integers with the sum of their reciprocals: n=(x+y+z)*(1/x+1/y+1/z).

Original entry on oeis.org

9, 10, 11, 14, 15, 18, 26, 30, 34, 35, 38, 42, 54, 55, 59, 62, 63, 70, 74, 82, 90, 95, 98, 102, 105, 122, 126, 131, 135, 138, 143, 158, 159, 170, 179, 190, 194, 195, 202, 203, 210, 215, 227, 230, 234, 238, 251, 255, 258, 266, 270, 278, 294, 297, 298, 310, 315

Offset: 1

Hugo Pfoertner, Jul 19 2003

nonn

All terms of this sequence occur also in A085514. Bremner et al. have shown that the problem is equivalent to finding rational points of infinite order on the elliptic curve E_n : u^2 = v^3 + (n^2 - 6*n - 3)*v^2 + 16*n*v

The only values of n < 1000 with positive representations are shown in bold type in Table 1 in Section 8 of Bremner et al.'s paper (except for the singular value n=9 and the case n=10) - Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 09 2008

			a(2)=(1+1+2)*(1/1+1/1+1/2)=10.
a(3)=(1+2+3)*(1/1+1/2+1/3)=6*(11/6)=11.
a(4)=(2+3+10)*(1/2+1/3+1/10)=14.
a(12)=(561+6450+13889)*(1/561+1/6450+1/13889)=42.

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.
A. MacLeod, The Knight's Problem
A. MacLeod, Elliptic Curves

Cf. A085514 (also negative x, y, z admitted).

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

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 09 2008

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A102774 Values of x arising from representations of n >= 11 in A085514.

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A102775 Values of y arising from representations of n >= 11 in A085514.

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A102777 Values of z arising from representations of n >= 11 in A085514.

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A102535 Integers n such that -n is 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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A309142 Rank of elliptic curve y^2 = x^3 + (n^2 - 6*n -3)*x^2 + 16*n*x.

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A086446 Integers representable as the product of the sum of three positive integers with the sum of their reciprocals: n=(x+y+z)*(1/x+1/y+1/z).

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A309142 Rank of elliptic curve y^2 = x^3 + (n^2 - 6n -3)x^2 + 16nx.