login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A103255 Integers x > 0 such that x^3 + y^3 = z^2 for some y > 0, z > 0, and gcd(x,y) = 1. 3
1, 2, 11, 23, 37, 56, 57, 65, 112, 122, 193, 217, 242, 305, 312, 433, 592, 781, 851, 877, 889, 913, 1001, 1064, 1177, 1201, 1346, 1376, 1617, 1633, 1706, 1729, 1801, 1953, 1960, 1969, 2137, 2162, 2184, 2257, 2345, 2480, 2543, 2920, 3071, 3081, 3482, 3641, 3889, 4019 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..50.

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

H. Darmon and A. Granville, On the equations z^m=F(x,y) and Ax^p+By^q=Cz^r, Bull. Lond. Math. Soc., 27 (6) (1995) 513, Sect 7.2.

EXAMPLE

x=11, y=37, 11^3 + 37^3 = 228^2. 11 is the third entry in the list.

The pairs [x,y] = [a(n),a(?)] for the first few terms are [1, 2], [2, 1], [11, 37], [23, 1177], [37, 11], [56, 65], [57, 112], [65, 56], [112, 57], [122, 1201], [193, 3482], [217, 312], [242, 433]. [Joerg Arndt, Sep 30 2012]

MATHEMATICA

(* This program uses z-values from A099426 b-file. To get 50 terms, the first 200 z-values suffice, the result being the same with the whole b-file of 300 z-values. *)

terms = 50;

zz = Import["https://oeis.org/A099426/b099426.txt", "Table"][[1 ;; 4 terms, 2]];

r[z_] := {x, y, z} /. ToRules[Reduce[GCD[x, y] == 1 && 0<x<y && x^3 + y^3 == z^2, {x, y}, Integers]];

xyz = r /@ zz;

Union[Flatten[xyz[[All, 1 ;; 2]]]][[1 ;; terms]] (* Jean-François Alcover, Jun 13 2019 *)

PROG

(MAGMA) [ k : k in [1..100] | exists{P : P in IntegralPoints(EllipticCurve([0, k^3])) | P[1] gt 0 and P[2] ne 0 and GCD(Integers()!P[1], k) eq 1} ]; // Geoff Bailey

(Sage) # apparently inefficient as of version 5.2

def is_A103255(n):

    E = EllipticCurve([0, n^3])

    E.gens(descent_second_limit=16);

    for p in E.integral_points():

        if p[0] > 0 and p[1] > 0 and gcd(p[1], n) == 1:

            return true

    return false

[n for n in (1..60) if is_A103255(n)]

# Peter Luschny, Sep 29 2012

(PARI)

is_A103255(x, lim)=

{ /* Warning: just how big lim has to be is unclear */

    my(x3=x^3);

    for (y=1, lim,

        if ( gcd(x, y) != 1, next() );

        if ( issquare(x3+y^3), return(1) );

    );

    return(0);

}

/* Using lim=10^6 reproduces all terms <= 1000: */

for (n=1, 1000, if( is_A103255(n, 10^6), print1(n, ", ")) );

/* Joerg Arndt, Sep 30 2012 */

CROSSREFS

Cf. A099426 (values of z).

Sequence in context: A085745 A106856 A045387 * A031385 A294551 A179878

Adjacent sequences:  A103252 A103253 A103254 * A103256 A103257 A103258

KEYWORD

nonn,more

AUTHOR

Cino Hilliard, Mar 20 2005

EXTENSIONS

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

a(9)-a(10) from Jonathan Vos Post, May 27 2007

a(11)-a(16) from Vincenzo Librandi, Dec 21 2010

a(17)-a(22) from Joerg Arndt, Sep 30 2012

a(23)-a(50) from Jean-François Alcover, Jun 12 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 12 07:28 EDT 2021. Contains 343821 sequences. (Running on oeis4.)