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A134108
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Number of integral solutions with nonnegative y to Mordell's equation y^2 = x^3 + n.
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7
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3, 1, 1, 1, 1, 0, 0, 4, 5, 1, 0, 2, 0, 0, 2, 1, 8, 1, 1, 0, 0, 1, 0, 4, 1, 1, 1, 2, 0, 1, 1, 0, 1, 0, 1, 4, 3, 1, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 3, 0, 0, 0, 0, 0, 2, 3, 4, 0, 0, 2, 0, 0, 1, 1, 6, 0, 0, 1, 0, 0, 1, 4, 1, 1, 0, 0, 0, 0, 0, 0, 4, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 6, 2, 0, 0, 0, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| a(n) = A081119(n)/2 if A081119(n) is even, (A081119(n)+1)/2 if A081119(n) is odd (i.e. if n is a cubic number).
Comment from T. D. Noe, Oct 12 2007: In sequences A134108 (this entry) and A134109 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 + 1 has solutions (y, x) = (0, -1), (1, 0) and (3, 2), hence a(1) = 3.
y^2 = x^3 + 6 has no solutions, hence a(6) = 0.
y^2 = x^3 + 17 has 8 solutions (see A029727, A029728), hence a(17) = 8.
y^2 = x^3 + 27 has solution (y, x) = (0, -3), hence a(27) = 1.
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PROG
| (MAGMA) [ #{ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, n])) }: n in [1..122] ];
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CROSSREFS
| Cf. A081119, A029727, A029728, A134109.
Sequence in context: A110245 A136093 A206831 * A176851 A205535 A172972
Adjacent sequences: A134105 A134106 A134107 * A134109 A134110 A134111
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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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