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A103254 Positive integers x such that there exist positive integers y and z satisfying x^3 + y^3 = z^2. 3
1, 2, 4, 7, 8, 9, 10, 11, 14, 16, 18, 21, 22, 23, 25, 26, 28, 32, 33, 34, 35, 36, 37, 38, 40, 44, 46, 49, 50, 56, 57, 63, 64, 65, 70, 72, 78, 81, 84, 86, 88, 90, 91, 92, 95, 98, 99, 100, 104, 105, 110, 112, 114, 121, 122, 126, 128, 129, 130, 132, 136, 140, 144, 148, 152, 154, 158, 160, 162, 169, 170, 175, 176, 177, 183, 184, 189, 190, 193, 196, 198, 200 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A001105 is a subset (excluding 0), since (x, y, z)=(A001105(k), A001105(k), A033430(k)) satisfies x^3+y^3=z^2. - R. J. Mathar, Sep 11 2006

A parametric solution: {x,y,z} = {g*(4*e + g)*(4*e^2 + 8*e*g + g^2), 2*g*(4*e + g)*(-2*e^2 +2*e*g + g^2), 3*g^2*(4*e + g)^2*(4*e^2 + 2*e*g + g^2)}, provided (-2*e^2 +2*e*g + g^2)>0. - James McLaughlin, Jan 27 2007

LINKS

Table of n, a(n) for n=1..82.

Fritz Beukers, The Diophantine equation Ax^p+By^q=Cz^r, Duke Math. J. 91 (1998), 61-88.

EXAMPLE

x=7, y=21, 7^3 + 21^3 = 98^2. 7 is the 4th term in the list.

Other solutions are (x, y, z)=(1, 2, 3), (4, 8, 24), (7, 21, 98), (9, 18, 81), (10, 65, 525), (11, 37, 228), (14, 70, 588), (16, 32, 192), (21, 7, 98), (22, 26, 168), (23, 1177, 40380), ...

PROG

(MAGMA) [ k : k in [1..200] | exists{P : P in IntegralPoints(EllipticCurve([0, k^3])) | P[1] gt 0 and P[2] ne 0 } ]; // Geoff Bailey, Jan 28 2007

CROSSREFS

See A103255 for another version.

Sequence in context: A285535 A342777 A173416 * A299419 A083454 A047542

Adjacent sequences:  A103251 A103252 A103253 * A103255 A103256 A103257

KEYWORD

nonn

AUTHOR

Cino Hilliard, Mar 20 2005

EXTENSIONS

Recomputed and extended to 48 terms by Geoff Bailey (geoff(AT)maths.usyd.edu.au) using MAGMA, Jan 28 2007

Terms 104..200 added by Joerg Arndt, Sep 29 2012

STATUS

approved

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Last modified May 17 22:15 EDT 2021. Contains 343992 sequences. (Running on oeis4.)