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 1-9 of 9 results.

A256418 Congrua (possible solutions to the congruum problem): numbers k such that there are integers x, y and z with k = x^2-y^2 = z^2-x^2.

Original entry on oeis.org

24, 96, 120, 216, 240, 336, 384, 480, 600, 720, 840, 864, 960, 1080, 1176, 1320, 1344, 1536, 1920, 1944, 2016, 2160, 2184, 2400, 2520, 2880, 2904, 3000, 3024, 3360, 3456, 3696, 3840, 3960, 4056, 4320, 4704, 4896, 5280, 5376, 5400, 5544
Offset: 1

Views

Author

N. J. A. Sloane, Apr 06 2015, following a suggestion from Robert Israel, Apr 03 2015

Keywords

Comments

k is a "congruum" iff k/4 is the area of a Pythagorean triangle, so these are the numbers 4*A009112.
Each congruum is a multiple of 24; it cannot be a square.
This entry incorporates many comments that were originally in A057102. A057103 and A055096 need to be checked.

Examples

			a(11)=840 since 840=29^2-1^2=41^2-29^2 (indeed also 840=37^2-23^2=47^2-37^2).

Links

Crossrefs

Cf. A004431 for possible values of x in definition. Cf. A057103, A055096 for triangles of all congrua and values of x.
Cf. A009112, A073120, A135789, A135786.

Programs

  • Mathematica
    r[n_] := Reduce[0 < y < x && 0 < x < z && n == x^2 - y^2 == z^2 - x^2, {x, y, z}, Integers];
Reap[For[n = 24, n < 10^4, n += 24, rn = r[n]; If[rn =!= False, Print[n, " ", rn]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 25 2019 *)

A057102 a(n) = 4 * A073120(n).

Original entry on oeis.org

24, 96, 120, 240, 336, 384, 480, 720, 840, 960, 1320, 1344, 1536, 1920, 1944, 2016, 2184, 2520, 2880, 3360, 3696, 3840, 3960, 4896, 5280, 5376, 5544, 6144, 6240, 6840, 6864, 7680, 7776, 8064, 8736, 9240, 9360, 9720, 10080, 10296, 10920, 11520, 12144
Offset: 1

Views

Author

Henry Bottomley, Aug 02 2000

Keywords

Comments

This sequence was originally described as the list of "congrua". But that name more properly refers to A256418.
Numbers of the form 4*(x^3*y-x*y^3) (where x,y are integers and x>=y). Squares of these numbers are of the form N^4-K^2 (where N belongs to A135786 and K to A135789 or A135790). Proof uses identity: (4(x^3y-xy^3))^2=(x^2+y^2)^4-(x^4 - 6x^2 y^2 + y^4)^2. - Artur Jasinski, Nov 29 2007, Nov 14 2008

Crossrefs

Cf. A073120, A256418.

Programs

  • Maple
    N:= 10^5: # to get all terms <= N
select(`<=`,{seq(seq(4*(x^3*y-x*y^3),y=1..x-1),x=1..floor(sqrt(N/4+1)))},N);
# If using Maple 11 or earlier, uncomment the following line
# sort(convert(%, list)); # Robert Israel, Apr 06 2015
  • Mathematica
    a = {}; Do[Do[w = 4x^3y - 4x y^3; If[w > 0 && w < 10000, AppendTo[a, w]], {x, y, 1000}], {y, 1, 1000}]; Union[a] (* Artur Jasinski, Nov 29 2007 *)

Extensions

Edited by N. J. A. Sloane, Apr 06 2015 at the suggestion of Robert Israel, Apr 03 2015

A135790 Positive numbers of the form -x^4+6x^2 y^2-y^4 (where x,y are integers).

Original entry on oeis.org

4, 7, 64, 112, 119, 164, 239, 324, 527, 567, 644, 959, 1024, 1519, 1792, 1904, 2047, 2500, 2624, 2884, 3479, 3824, 4207, 4324, 4375, 4879, 4964, 5184, 5572, 6647, 6887, 7327, 8119, 8432, 9072, 9604, 9639
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 A057102). Proof uses: (x^4 - 6x^2 y^2 + y^4)^2=(x^2+y^2)^4-(4(x^3y-xy^2))^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.

Programs

  • Mathematica
    a = {}; Do[Do[w = -x^4 + 6x^2 y^2 - y^4; If[w > 0&&w<10000, AppendTo[a, w]], {x, y, 2000}], {y, 1, 2000}]; Union[a]

A135791 Positive numbers of the form x^5-10x^3*y^2+5x*y^4 (where x,y are integers and x>y).

Original entry on oeis.org

404, 1900, 3647, 5646, 12928, 13412, 14050, 27688, 30609, 36413, 45716, 51804, 60800, 74576, 90050, 98172
Offset: 1

Views

Author

Artur Jasinski, Nov 29 2007

Keywords

Comments

See A135792, union A135791 and A135792 see A135793. Squares of these numbers are of the form N^5-M^2 (where N belongs to A135787 and M to A057102) Proof uses: (x^5-10x^3 y^2+5xy^4)^2=(x^2+y^2)^5-(5x^4y-10x^2y^3+y^5)^2. [This line needs editing! - N. J. A. Sloane, Dec 04 2007]
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, A135792, A135793.

Programs

  • Mathematica
    a = {}; Do[Do[w = x^5 - 10x^3 y^2 + 5x y^4; If[w > 0 && w < 100000, AppendTo[a, w]], {x, y, 1000}], {y, 1, 1000}]; Union[a]

A135792 Positive numbers of the form x^5-10x^3*y^2+5x*y^4 (where x,y are integers and y>x).

Original entry on oeis.org

41, 122, 316, 1121, 1312, 1900, 2868, 2876, 3904, 4282, 6121, 9963, 10112, 11516, 17684, 19841, 20122, 23028, 23807, 25525, 29646, 31996, 35872, 41984, 44403, 49001, 59162, 60800, 65900, 71996, 76453, 76788, 80404, 91776, 92032
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, A135793.

Programs

  • Mathematica
    a = {}; Do[Do[w = x^5 - 10x^3 y^2 + 5x y^4; If[w > 0 && w < 100000, AppendTo[a, w]], {y, x, 1000}], {x, 1, 1000}]; Union[a]

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.

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]
Showing 1-9 of 9 results.

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

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