A309144 -id:A309144 - cp's OEIS frontend

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).

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]}

A309147 Least k such that the rank of the elliptic curve y^2 = x^3 + (k^2 + 6k - 3)x^2 - 16kx is n.

Original entry on oeis.org

1, 4, 28, 356

Offset: 0

Seiichi Manyama, Jul 14 2019

Allan J. MacLeod, Knight's Problem

Cf. A309144, A309146.

{a(n) = my(k=1); while(ellanalyticrank(ellinit([0, k^2+6*k-3, 0, -16*k, 0]))[1]<>n, k++); k}

A309145 Numbers k for which rank of the elliptic curve y^2=x^3+(k^2+6k-3)x^2-16kx is 2.

Original entry on oeis.org

28, 52, 59, 70, 76, 101, 103, 108, 115, 122, 130, 139, 148, 164, 172, 180, 181, 190, 199, 208, 210, 220, 222, 223, 228, 268, 270, 284, 314, 316, 327, 328, 339, 340, 364, 376, 388, 398, 403, 420, 427, 430, 436, 443, 446, 448, 456, 457, 460, 480, 487, 490, 504, 521, 532, 540

Offset: 1

Seiichi Manyama, Jul 14 2019

nonn

Allan J. MacLeod, Knight's Problem

Cf. A309144.

for(k=1, 1e3, if(ellanalyticrank(ellinit([0, k^2+6*k-3, 0, -16*k, 0]))[1]==2, print1(k", ")))

A309144(a(n)) = 2.

cp's OEIS Frontend

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 - 6n -3)x^2 + 16nx.

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A309147 Least k such that the rank of the elliptic curve y^2 = x^3 + (k^2 + 6k - 3)x^2 - 16kx is n.

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A309145 Numbers k for which rank of the elliptic curve y^2=x^3+(k^2+6k-3)x^2-16kx is 2.

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cp's OEIS Frontend

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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Author

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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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PARI

A309147 Least k such that the rank of the elliptic curve y^2 = x^3 + (k^2 + 6*k - 3)*x^2 - 16*k*x is n.

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Author

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PARI

A309145 Numbers k for which rank of the elliptic curve y^2=x^3+(k^2+6*k-3)*x^2-16*k*x is 2.

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Author

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Formula

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

A309147 Least k such that the rank of the elliptic curve y^2 = x^3 + (k^2 + 6k - 3)x^2 - 16kx is n.

A309145 Numbers k for which rank of the elliptic curve y^2=x^3+(k^2+6k-3)x^2-16kx is 2.