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 11-15 of 15 results.

A135793 Numbers of the form x^5-10x^3*y^2+5x*y^4 (where x,y are integers).

Original entry on oeis.org

41, 122, 316, 404, 1121, 1312, 1900, 2868, 2876, 3647, 3904, 4282, 5646, 6121, 9963, 10112, 11516, 12928, 13412, 14050, 17684, 19841, 20122, 23028, 23807, 25525, 27688, 29646, 30609, 31996, 35872, 36413, 41984, 44403, 45716, 49001, 51804
Offset: 1

Views

Author

Artur Jasinski, Nov 29 2007

Keywords

Comments

Refers to A057102, which had an incorrect description and has been replaced by A256418. As a result the present sequence should be re-checked. - N. J. A. Sloane, Apr 06 2015

Crossrefs

Cf. A000404, A050803, A057102, A135784, A060803, A135786, A135787, A135789, A135790, A135791, A135792.

A050804 Numbers n such that n^3 is the sum of two nonzero squares in exactly one way.

Original entry on oeis.org

2, 8, 18, 32, 72, 98, 128, 162, 242, 288, 392, 512, 648, 722, 882, 968, 1058, 1152, 1458, 1568, 1922, 2048, 2178, 2592, 2888, 3528, 3698, 3872, 4232, 4418, 4608, 4802, 5832, 6272, 6498, 6962, 7688, 7938, 8192
Offset: 1

Views

Author

Patrick De Geest, Sep 15 1999

Keywords

Comments

m is a term if and only if m = 2^(2a_0+1)*p_1^(2a_1)*p_2^(2a_2)*...*p_k^(2a_k), where a_i are nonnegative integers and p_i are primes of the form 4k+3. - Chai Wah Wu, Feb 27 2016
m is a term if and only if for all odd q > 1, m^q is the sum of two nonzero squares in exactly one way. - Chai Wah Wu, Feb 28 2016
Numbers n such that n is the sum of two nonzero squares while n^2 is not. - Altug Alkan, Apr 11 2016

Examples

			E.g. 32^3 = 128^2 + 128^2. Is there an example using different squares?
No: If n^3 has only one representation as n^3 = a^2+b^2 with 0<a<=b, then a=b. - _Jonathan Vos Post_, Feb 02 2011

Links

Crossrefs

Cf. A000404, A050803.
Cf. A081324.

Programs

  • Haskell
    a050804 n = a050804_list !! (n-1)
a050804_list = filter ((== 1) . a084888) [0..]
-- Reinhard Zumkeller, Jul 18 2012
  • Mathematica
    ok[n_] := Length @ Cases[ PowersRepresentations[n^3, 2, 2], {?Positive, ?Positive}] == 1; Select[Range[8200], ok] (* Jean-François Alcover, Apr 05 2011 *)
  • Python
    from sympy import factorint
A050804_list = [2*i for i in range(1,10**6) if not any(p % 4 == 1 or factorint(i)[p] % 2 for p in factorint(i))] # Chai Wah Wu, Feb 27 2016

Formula

n such that A084888(n) = 1.

Extensions

More terms from Michel ten Voorde and Jud McCranie

A135796 Numbers of the form 4x^3y+4y x^3 (where x,y are positive integers).

Original entry on oeis.org

8, 40, 120, 128, 272, 312, 520, 640, 648, 888, 1160, 1200, 1400, 1920, 2040, 2048, 2080, 2952, 2968, 3240, 3280, 4040, 4352, 4872, 4992, 5000, 5368, 6120, 6960, 7008, 7280, 7320, 8320, 8840, 9720
Offset: 1

Views

Author

Artur Jasinski, Nov 29 2007

Keywords

Comments

Squares of these numbers are of the form N^4-M^2 (where N belongs to A135786 and M to A135797) Proof uses: (4x^3y+4xy^3)^2=(x^2-y^2)^4+(x^4+6x^2y^2+y^4)^2.
Refers to A057102, which had an incorrect description and has been replaced by A256418. As a result the present sequence should be re-checked. - N. J. A. Sloane, Apr 06 2015

Crossrefs

Cf. A000404, A050803, A057102, A135784, A060803, A135786, A135787, A135789, A135790, A135791, A135792, A135793, A135795, A135796, A135797.

Programs

  • Mathematica
    a = {}; Do[Do[w = 4x^3y + 4x y^3; If[w < 10000, AppendTo[a, w]], {x, y, 1000}], {y, 1, 1000}]; Union[a] (*Artur Jasinski*)

A135797 Numbers of the form x^4 + 6*x^2*y^2 + y^4 (where x,y are positive integers).

Original entry on oeis.org

8, 41, 128, 136, 313, 353, 648, 656, 776, 1201, 1241, 1513, 2048, 2056, 2176, 2696, 3281, 3321, 3593, 4481, 5000, 5008, 5128, 5648, 7048, 7321, 7361, 7633, 8521
Offset: 1

Views

Author

Artur Jasinski, Nov 29 2007

Keywords

Comments

Squares of these numbers are of the form N^4-M^2 (where N belongs to A135786 and M to A135796). Proof uses: (x^4+6*x^2*y^2+y^4)^2 = (x^2-y^2)^4+(4*x^3*y+4*x*y^3)^2.
Refers to A057102, which had an incorrect description and has been replaced by A256418. As a result the present sequence should be re-checked. - N. J. A. Sloane, Apr 06 2015

Crossrefs

Cf. A000404, A050803, A057102, A135784, A060803, A135786, A135787, A135789, A135790, A135791, A135792, A135793, A135795, A135796, A135796.

Programs

  • Mathematica
    a = {}; Do[Do[w = x^4 + 6x^2 y^2 + y^4; If[w < 10000, AppendTo[a, w]], {x, y, 1000}], {y, 1, 1000}]; Union[a]
(* alternate program *)
Union[Select[#[[1]]^4+6#[[1]]^2 #[[2]]^2+#[[2]]^4&/@Tuples[Range[ 1000],2], #<10000&]] (* Harvey P. Dale, Oct 07 2012 *)

A135794 Numbers of the form x^5 + 10*x^3*y^2 + 5*x*y^4 (where x,y are integers).

Original entry on oeis.org

16, 121, 122, 496, 512, 528, 1441, 1562, 1563, 1684, 3376, 3872, 3888, 3904, 4400, 6841, 8282, 8403, 8404, 8525, 9966, 12496, 15872, 16368, 16384, 16400, 16896, 20272, 21121, 27962, 29403, 29524, 29525, 29646, 31087, 33616, 37928, 46112
Offset: 1

Views

Author

Artur Jasinski, Nov 29 2007, Oct 10 2008

Keywords

Comments

Squares of these numbers are of the form N^5+M^2 (where N belongs to A000404 and M to A135795). Proof uses: (x^5+10x^3 y^2+5xy^4)^2=(x^2-y^2)^5+(5x^4y+10x^2y^3+y^5)^2.
Also numbers of the form ((y + x)^5 - (y - x)^5)/2 = x^5 + 10*x^3*y^2 + 5*x*y^4. - Artur Jasinski, Oct 10 2008
Refers to A057102, which had an incorrect description and has been replaced by A256418. As a result the present sequence should be re-checked. - N. J. A. Sloane, Apr 06 2015

Crossrefs

Cf. A000404, A050803, A057102, A135784, A060803, A135786, A135787, A135789, A135790, A135791, A135792, A135793, A135795, A135796.

Programs

  • Mathematica
    a = {}; Do[Do[w = x^5 + 10x^3 y^2 + 5x y^4; If[w < 100000, AppendTo[a, w]], {x, 1, 1000}], {y, 1, 1000}]; Union[a]
Previous Showing 11-15 of 15 results.

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

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