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A134109
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Number of integral solutions with nonnegative y to Mordell's equation y^2 = x^3 - n.
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2
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1, 1, 0, 2, 0, 0, 2, 1, 0, 0, 2, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 2, 1, 3, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 1, 0, 0, 0, 1, 1, 0, 3, 2, 1, 0, 0, 0, 2, 1, 2, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 3
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| a(n) = A081120(n)/2 if A081120(n) is even, (A081120(n)+1)/2 if A081120(n) is odd (i.e. if n is a cubic number).
Comment from T. D. Noe, Oct 12 2007: In sequences A134108 and A134109 (this entry) dealing with the equation y^2 = x^3 + n, one could note that these are Mordell equations. Here are some related sequences: A054504, A081119, A081120, A081121. The link "Integer points on Mordell curves" has data on 20000 values of n. A134108 and A134109 count only solutions with y >= 0 and can be derived from A081119 and A081120.
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LINKS
| J. Gebel, Integer points on Mordell curves
Eric Weisstein's World of Mathematics, Mordell Curve
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EXAMPLE
| y^2 = x^3 - 4 has solutions (y, x) = (2, 2) and (11, 5), hence a(4) = 2.
y^2 = x^3 - 5 has no solutions, hence a(5) = 0.
y^2 = x^3 - 8 has solution (y, x) = (0, 2), hence a(8) = 1.
y^2 = x^3 - 207 has 7 solutions (see A134106, A134107), hence a(207) = 7.
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PROG
| (MAGMA) [ #{ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, -n])) }: n in [1..104] ];
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CROSSREFS
| Cf. A081120, A134106, A134107, A134108.
Sequence in context: A093693 A025436 A170976 * A170978 A161364 A143620
Adjacent sequences: A134106 A134107 A134108 * A134110 A134111 A134112
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KEYWORD
| nonn
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AUTHOR
| Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Oct 08 2007, Oct 14 2007
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