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.

Previous Showing 11-20 of 29 results. Next

A228948 Numbers n such that n^3 + k^2 = m^3 for some k>0, m>0.

Original entry on oeis.org

6, 7, 11, 23, 24, 26, 28, 31, 38, 42, 44, 47, 54, 55, 61, 63, 84, 91, 92, 95, 96, 99, 104, 110, 111, 112, 118, 119, 124, 138
Offset: 1

Views

Author

M. F. Hasler, Oct 05 2013

Keywords

Comments

Cube root of perfect cubes in A087285 or in A229618 are in the present sequence, but this does not yield all terms, because these sequences require k^2 to be the largest square < m^3.
Numbers k such that Mordell's equation y^2 = x^3 - k^3 has more than 1 integral solution. (Note that it is necessary that x is positive.) In other words, numbers k such that Mordell's equation y^2 = x^3 - k^3 has solutions other than the trivial solution (k,0). - Jianing Song, Sep 24 2022

Examples

			6 is a term since the equation y^2 = x^3 - 6^3 has 5 solutions (6,0), (10,+-28), and (33,+-189). - _Jianing Song_, Sep 24 2022

Crossrefs

Cf. A229618, A087285, A087286, A088017, A081121, A081120, A077116, A065733.
Cube root of A179419.
Cf. A356709, A356720. Complement of A356713.

Extensions

More terms added by Jianing Song, Sep 24 2022 based on A179419.

A110223 Numbers not the absolute difference between a cube and a square.

Original entry on oeis.org

6, 14, 21, 29, 32, 34, 42, 46, 51, 58, 59, 62, 66, 69, 70, 75, 77, 78, 84, 85, 86, 88, 90, 93, 96, 102, 103, 110, 111, 114, 115, 123, 130, 133, 137, 140, 149, 157, 158, 160, 162, 165, 166, 173, 176, 178, 179, 181, 182, 183, 187, 194, 201, 202, 203, 205, 209, 210, 211
Offset: 1

Views

Author

Robert G. Wilson v, Jul 16 2005

Keywords

Comments

See A074981 for references.

Links

Crossrefs

Cf. A074981. Intersection of A081121 and A054504.

Programs

  • Mathematica
    Complement[ Range[212], Union[ Flatten[ Table[ Select[ Table[ Abs[n^3 - m^2], {m, 0, 10000}], # < 10^3 &], {n, -5000, 5000}]]]]

A177986 Numbers n such that quartic curve y^2=x^4+n have integral points.

Original entry on oeis.org

1, 3, 4, 8, 9, 15, 16, 19, 20, 24, 25, 28, 33, 35, 36, 40, 48, 49, 51, 63, 64, 65, 68, 73, 80, 81, 84, 99, 100, 104
Offset: 1

Views

Author

Artur Jasinski, May 16 2010

Keywords

Comments

To this sequence belonging as subset all perfect squares and squares-1.
Complement to this sequence see A177987.

Crossrefs

Cf. A054504, A081121.

A177987 Numbers n such that quartic equation y^2=x^4+n has no solution.

Original entry on oeis.org

2, 5, 6, 7, 10, 11, 12, 13, 14, 17, 18, 21, 22, 23, 26, 27, 29, 30, 31, 32, 34, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 66, 67, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98
Offset: 1

Views

Author

Artur Jasinski, May 16 2010

Keywords

Comments

Complement to this sequence see A177986.

Examples

			104 does not belong to this sequence because 27^2 = 5^4 + 104.

Crossrefs

Cf. A054504, A081121.

A125643 Squares and cubes (with repetition).

Original entry on oeis.org

0, 0, 1, 1, 4, 8, 9, 16, 25, 27, 36, 49, 64, 64, 81, 100, 121, 125, 144, 169, 196, 216, 225, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 729, 784, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681
Offset: 1

Views

Author

Zak Seidov, Oct 19 2006

Keywords

Comments

Repeating terms are sixth powers: 0,1,64,729,... (A001014).
For numbers not appearing as a difference between a square and an adjacent cube in this list, see A054504 and A081121.

Links

Crossrefs

Cf. A002760 (squares and cubes (without repetitions)).
Cf. A001014, A054504, A081121, A087285, A087286, A088017.

Programs

  • Mathematica
    m=1681;cm=Floor[m^(1/3)];sm=Floor[Sqrt[m]];s=Range[0,sm]^2;c=Range[0,cm]^3;Sort[Join[s,c]] (* James C. McMahon, Dec 20 2024 *)
  • Python
    from math import isqrt
from sympy import integer_nthroot
def A125643(n):
    if n <= 4: return n-1>>1
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n-2+x-integer_nthroot(x,3)[0]-isqrt(x)
    return bisection(f,n-2,n-2) # Chai Wah Wu, Oct 14 2024

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jul 14 2007

A177988 Numbers n such that quartic curve y^2=x^4-n has integral points.

Original entry on oeis.org

1, 7, 12, 15, 16, 17, 31, 32, 45, 49, 56, 60, 65, 71, 72, 77, 80, 81
Offset: 1

Views

Author

Artur Jasinski, May 16 2010

Keywords

Comments

Complement to this sequence see A177989.
Numbers n such that quartic curve y^2=x^4+n has integral points. see A177986.

Crossrefs

Cf. A054504, A081121, A177986, A177987, A177988, A177989.

Programs

  • Magma
    IntegralQuarticPoints([1,0,0,0,-81]);

A177989 Numbers n such that quartic equation y^2=x^4-n has no integer solution.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 14, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 50, 51, 52, 53, 54, 55, 57, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 70, 73, 74, 75, 76, 78, 79
Offset: 1

Views

Author

Artur Jasinski, May 16 2010

Keywords

Comments

Complement to this sequence see A177989.
Numbers n such that the quartic curve y^2=x^4+n doesn't have integral points. see A177987.

Crossrefs

Cf. A054504, A081121, A177986, A177987, A177988, A177989.

Programs

  • Magma
    IntegralQuarticPoints([1,0,0,0,-79]);

A179174 Numbers n such that Mordell's equation y^2 = x^3 - n has exactly 22 integral solutions.

Original entry on oeis.org

3807, 3896, 52784, 129556, 157239, 167600, 185112, 200871, 281439, 314199, 347967, 370647, 399375, 553648, 623872, 720703, 815728, 819775, 856799, 934975, 994816
Offset: 1

Views

Author

Artur Jasinski, Jun 30 2010

Keywords

Comments

Counting (+x,+y) and (+x,-y) iff y != 0.

Links

Crossrefs

Subsequence of A007412.
Cf. A081120, A081121, A179163-A179175, A134107, A106265.

Extensions

Edited by Ray Chandler, Jul 11 2010
a(3)-a(21) from Jose Aranda, Aug 10 2024

A154332 Least positive integer m such that A087285(n) = A154333(m) = m^3 - next smaller square.

Original entry on oeis.org

3, 2, 32, 15, 17, 4, 7, 6, 35, 8, 11, 10, 14, 21, 12, 28, 65, 9, 56, 18, 136, 568, 23, 99, 101, 20, 13, 27, 34, 30, 143, 145, 38, 16, 19, 47, 195, 91, 197, 175, 26, 51, 59, 799, 69, 62, 163, 255, 257, 66, 31, 717, 2904, 33, 377, 79, 323, 325, 25
Offset: 1

Views

Author

M. F. Hasler, Jan 07 2009

Keywords

Comments

The terms of this sequence constitute a "proof" for the terms listed in A087285. To prove that a number is NOT in A087285, one can check the finite number (A081120) of solutions to the corresponding Mordell equation, cf. references in A081121.

Programs

  • PARI
    A154332(n) = { local(m); until(m++^3-sqrtint(m^3-1)^2==A087285[n],); m }

Formula

A087285(n) = A154333(a(n)) = a(n)^3 - [sqrt(a(n)^3 - 1)]^2 = A000578(a(n)) - A048760(a(n)^3-1).

A179164 Numbers n such that Mordell's equation y^2 = x^3 - n has exactly 2 integral solutions.

Original entry on oeis.org

2, 13, 15, 18, 19, 20, 23, 25, 35, 40, 44, 45, 49, 54, 56, 61, 67, 71, 72, 74, 79, 81, 83, 87, 89, 95, 106, 107, 112, 118, 121, 124, 126, 127, 128, 139, 143, 146, 148, 150, 151, 153, 155, 159, 167, 170, 172, 175, 184, 186, 188, 193, 199, 222, 223, 233, 235, 236, 239
Offset: 1

Views

Author

Artur Jasinski, Jun 30 2010

Keywords

Links

Crossrefs

Cf. A081120, A081121, A179163-A179175.

Extensions

Edited by Ray Chandler, Jul 11 2010
Previous Showing 11-20 of 29 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.