|
Search: id:A134109
|
|
|
| A134109 |
|
Number of integral solutions with nonnegative y to Mordell's equation y^2 = x^3 - n. |
|
+0
2
|
|
| 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; listen)
|
|
|
OFFSET
|
1,4
|
|
|
COMMENT
|
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.
|
|
LINKS
|
J. Gebel, Integer points on Mordell curves
Eric Weisstein's World of Mathematics, Mordell Curve
|
|
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.
|
|
PROGRAM
|
(MAGMA) [ #{ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, -n])) }: n in [1..104] ];
|
|
CROSSREFS
|
Cf. A081120, A134106, A134107, A134108.
Sequence in context: A079181 A093693 A025436 this_sequence A143620 A025843 A035437
Adjacent sequences: A134106 A134107 A134108 this_sequence A134110 A134111 A134112
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Oct 08 2007, Oct 14 2007
|
|
|
Search completed in 0.002 seconds
|
