cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A099426 Numbers n where n^2 = x^3 + y^3; x,y>0 and gcd(x,y)=1.

Original entry on oeis.org

3, 228, 671, 1261, 6371, 9765, 35113, 35928, 40380, 41643, 66599, 112245, 124501, 127499, 167160, 191771, 205485, 255720, 297037, 377567, 532392, 546013, 647569, 681285, 812340, 897623, 1043469, 1125683, 1261491, 1431793, 1433040, 1584828, 1783067, 1984009, 2107391, 2372903, 2440893, 2484469, 2548557
Offset: 1

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Author

Hans Havermann, Oct 15 2004

Keywords

Comments

Based on an observation of Ed Pegg Jr, who supplied terms a(2)-a(6) and a(8).

Examples

			228 is in the sequence because 228^2 = 11^3 + 37^3 and gcd(11, 37) = 1.

Links

Crossrefs

Cf. A099532, A099533, A103255 (min(x,y), sorted), A282639 (max(x,y)).

Programs

  • Mathematica
    n = 10^7; n2 = n^2; x = 1; x3 = x^3; Reap[ While[x3 < n2, y = x + 1; y3 = y^3; While[y3 < n2, If[GCD[x, y] == 1, s = x3 + y3; If[ IntegerQ[r = Sqrt[s]], Print[r]; Sow[r]; Break[]]]; y += 1; y3 = y^3]; x += 1; x3 = x^3]][[2, 1]] // Sort (* Jean-François Alcover, Jan 11 2013, translated from Joerg Arndt's 2nd Pari program *)
  • PARI
    is_A099426(n)=
{
    my(n2=n^2, k=1, k3=1, r);
    while( k3 < n2,
        if ( ispower(n2-k3, 3, &r),
            if ( gcd(r,k)==1, return(1) );
        );
        k+=1;  k3=k^3;
    );
    return(0);
}
for (n=1,10^8, if( is_A099426(n), print1(n,", ")) );
/* Joerg Arndt, Sep 30 2012 */
  • PARI
    /* compute all terms below a threshold at once, terms need to be sorted */
{ N = 10^7; N2 = N^2;
x=1; x3=x^3;
while ( x3 < N2,
    y=x+1; y3=y^3;
    while ( y3 < N2,
        if ( gcd(x,y) == 1,
            s = x3 + y3;
            if ( issquare(s, &r), print(r); break(); );
        );
        y+=1;  y3 = y^3;
    );
    x+=1;  x3 = x^3;
);}
/* Joerg Arndt, Sep 30 2012 */
  • PARI
    for(s=2,1e5,for(x=1,s\2,my(y=s-x);if(gcd(x,y)>1,next); if(issquare(x^3+y^3), print1(s", ")))) \\ Charles R Greathouse IV, Nov 06 2014

Extensions

More terms from Hans Havermann and Bodo Zinser, Oct 20 2004

A103254 Positive integers x such that there exist positive integers y and z satisfying x^3 + y^3 = z^2.

Original entry on oeis.org

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
Offset: 1

Views

Author

Cino Hilliard, Mar 20 2005

Keywords

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 Mc Laughlin, Jan 27 2007
Allowing y = 0 would give the same sequence, since x^3 = z^2 implies x is a square, and all squares are terms since (t^2)^3 + (2*t^2)^3 = (3*t^3)^2. On the other hand, allowing y to be negative would introduce new terms: 71, 74, and 155 would be terms since 71^3 + (-23)^3 = 588^2, 74^3 + (-47)^3 = 549^2, and 155^3 + (-31)^3 = 1922^2. See A356720. - Jianing Song, Aug 24 2022

Examples

			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), ...

Links

Crossrefs

See A103255 for another version.

Programs

  • 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

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

A121980 Positive integers z, without duplication, in x^3+y^3=z^2.

Original entry on oeis.org

1, 3, 4, 8, 13, 24, 27, 28, 32, 49, 64, 81, 98, 104, 108, 125, 147, 168, 181, 189, 192, 216, 224, 228, 256, 312, 343, 351, 361, 375, 388, 392, 500, 507, 512, 525, 549, 588, 648, 671, 676, 729, 756, 784, 832, 847, 864, 1000, 1014, 1029, 1176, 1183, 1225, 1261
Offset: 1

Views

Author

R. J. Mathar, Sep 11 2006

Keywords

Comments

The first duplicate is (-23,71,588),(14,70,588), the second (-119,140,1029),(49,98,1029). A033430(m) and A000578(k) are subsets since (x,y,z)=(2m,2m,4m^3) or (x,y,z)=(0,k^2,k^3) solve x^3+y^3=z^2. The "leakage" problem of A103254 can be avoided by introducing s=x+y and d=y-x and searching for solutions of the transformed equation s(s^2+3d^2)=4z^2 over all positive divisors s of 4z^2.

Examples

			(x,y,z)=(0,1,1),(1,2,3),(2,2,4),(0,4,8),(-7,8,13),(4,8,24),(0,9,27),(-6,10,28),
(8,8,32),(-7,14,49),(0,16,64),(9,18,81),(7,21,98),(-28,32,104).

Crossrefs

Cf. A103254, A103255.
Showing 1-3 of 3 results.

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