A057102 -id:A057102 - cp's OEIS frontend

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.

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

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

nonn

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.

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

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Congruum (but beware errors)
Wikipedia, Congruum (but beware errors).

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.

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 *)

A073120 Areas of Pythagorean (or right) triangles with integer sides of the form (2mn, m^2 - n^2, m^2 + n^2).

Original entry on oeis.org

6, 24, 30, 60, 84, 96, 120, 180, 210, 240, 330, 336, 384, 480, 486, 504, 546, 630, 720, 840, 924, 960, 990, 1224, 1320, 1344, 1386, 1536, 1560, 1710, 1716, 1920, 1944, 2016, 2184, 2310, 2340, 2430, 2520, 2574, 2730, 2880, 3036, 3360, 3570, 3696, 3750, 3840

Offset: 1

Zak Seidov, Aug 25 2002

Equivalently, integers of the form m*n*(m^2 - n^2) where m,n are positive integers with m > n. - James R. Buddenhagen, Aug 10 2008
The sequence giving the areas of all Pythagorean triangles is A009112 (sometimes called "Pythagorean numbers").
For example, the sequence does not contain 54, the area of the Pythagorean triangle with sides (9,12,15). - Robert Israel, Apr 03 2015
See also Theorem 2 of Mohanty and Mohanty. - T. D. Noe, Sep 24 2013

			6 = 3*4/2 is the area of the right triangle with sides 3 and 4.
84 = 7*24/2 is the area of the right triangle with sides 7 and 24.

T. D. Noe, Table of n, a(n) for n = 1..10000
Supriya Mohanty and S. P. Mohanty, Pythagorean Numbers, Fibonacci Quarterly 28 (1990), 31-42.
Eric Weisstein's World of Mathematics, Pythagorean Triple.
Konstantine Hermes Zelator, A little noticed right triangle, arXiv:0804.1340 [math.GM], 2008.

Cf. A009112, A002144, A003273, A046081, A057102, A024365.

nn = 16; t = Union[Flatten[Table[m*n*(m^2 - n^2), {m, 2, nn}, {n, m - 1}]]]; Select[t, # < nn*(nn^2 - 1) &]

a(n) = A057102(n) / 4. - Max Alekseyev, Nov 14 2008

Description corrected by James R. Buddenhagen, Aug 10 2008, and by Max Alekseyev, Nov 12 2008

Edited by N. J. A. Sloane, Apr 06 2015

A135789 Positive numbers of the form x^4 - 6 * x^2 * y^2 + y^4 (where x,y are integers).

Original entry on oeis.org

28, 41, 161, 448, 476, 656, 721, 956, 1081, 1241, 1393, 2108, 2268, 2576, 3281, 3321, 3713, 3836, 4633, 4681, 5593, 6076, 7168, 7616, 8188, 9401, 9641, 10496, 11536, 11753, 12121, 12593, 13041, 13916, 15296, 16828, 17296, 17500, 19516, 19856

Offset: 1

Artur Jasinski, Nov 29 2007, Nov 14 2008

nonn

Squares of these numbers are of the form N^4 - M^2 (where N belongs to A135786 and M to A057102). Proof is based on the identity (x^4 - 6x^2 * y^2 + y^4)^2 = (x^2 + y^2)^4 - (4(x^3y - xy^3))^2.
Since x^4 - 6x^2 * y^2 + y^4 = d*d' where d = x^2 - y^2 + 2xy and d' = x^2 - y^2 - 2xy, and d - d' = 4xy, the computational technique is to consider the divisors d|n, d'=n/d, to check that the difference is a multiple of 4, and to check x in the range 1..d/3. - R. J. Mathar, Sep 18 2009
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

Cf. A000404, A050803, A057102, A135784, A060803, A135786, A135787.

isA135789 := proc(n) for d in numtheory[divisors](n) do dprime := n/d ; if abs(d-dprime) mod 4 = 0 then for x from 1 to d/3 do y := (d-dprime)/4/x ; if type(y,'integer') and y< x and y> 0 then if n = (x^2-y^2+2*x*y)*(x^2-y^2-2*x*y) then RETURN(true); fi; fi; od: fi: od: RETURN(false) ; end: for n from 1 do if isA135789(n) then printf("%d,\n",n) ; fi; od: # R. J. Mathar, Sep 18 2009

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]

More terms from R. J. Mathar, Sep 18 2009

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

Artur Jasinski, Nov 29 2007

nonn

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

Cf. A000404, A050803, A057102, A135784, A060803, A135786, A135787, A135789.

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^3y^2+5xy^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

Artur Jasinski, Nov 29 2007

nonn

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

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

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^3y^2+5xy^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

Artur Jasinski, Nov 29 2007

nonn

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

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

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^3y^2+5xy^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

Artur Jasinski, Nov 29 2007

nonn

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

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

A057103 Triangle of congrua: T(n,k) = 4nk(n^2-k^2) with n>k>0 and starting at T(2,1) = 24. A055096(n)^2 + a(n) is a square, as is A055096(n)^2 - a(n).

Original entry on oeis.org

24, 96, 120, 240, 384, 336, 480, 840, 960, 720, 840, 1536, 1944, 1920, 1320, 1344, 2520, 3360, 3696, 3360, 2184, 2016, 3840, 5280, 6144, 6240, 5376, 3360, 2880, 5544, 7776, 9360, 10080, 9720, 8064, 4896, 3960, 7680, 10920, 13440, 15000, 15360, 14280, 11520, 6840

Offset: 2

Henry Bottomley, Aug 02 2000

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

			T(2,1) = 4*2*1*(4-1) = 5^2-1^2 = 7^2-5^2 = 24.
Triangle begins:
   24;
   96,  120;
  240,  384,  336;
  480,  840,  960,  720;
  840, 1536, 1944, 1920, 1320;
  ...

Eric Weisstein's World of Mathematics, Congruum Problem.

Cf. possible congrua A057102. See also A055096.

T[n_, k_] := 4 n k (n^2 - k^2);
Table[T[n, k], {n, 2, 10}, {k, 1, n - 1}] // Flatten (* Jean-François Alcover, Feb 25 2019 *)

Offset corrected by Alois P. Heinz, Feb 25 2019

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

Artur Jasinski, Nov 29 2007

nonn

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

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

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*)

A056203 Triangle of numbers related to congruum problem: T(n,k)=n^2+2nk-k^2 with n>k>0, starting at T(2,1)=7.

Original entry on oeis.org

7, 14, 17, 23, 28, 31, 34, 41, 46, 49, 47, 56, 63, 68, 71, 62, 73, 82, 89, 94, 97, 79, 92, 103, 112, 119, 124, 127, 98, 113, 126, 137, 146, 153, 158, 161, 119, 136, 151, 164, 175, 184, 191, 196, 199, 142, 161, 178, 193, 206, 217, 226, 233, 238, 241, 167, 188

Offset: 1

Henry Bottomley, Aug 02 2000

The congruum problem is to find h (A057103) such that there are integers x (A055096), y (A057105) and z (A056203) with h=x^2-y^2=z^2-x^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

			a(1) = T(2,1) = 2^2+2*2*1-1 = 7.

Eric Weisstein's World of Mathematics, Congruum Problem.

Cf. A057102.

a(n) = sqrt(A057103(n)+A055096(n)^2) = sqrt(2*A057103(n)+A057105(n)^2).

cp's OEIS Frontend

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.

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A073120 Areas of Pythagorean (or right) triangles with integer sides of the form (2mn, m^2 - n^2, m^2 + n^2).

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A135789 Positive numbers of the form x^4 - 6 * x^2 * y^2 + y^4 (where x,y are integers).

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

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

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

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

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A057103 Triangle of congrua: T(n,k) = 4*n*k(n^2-k^2) with n>k>0 and starting at T(2,1) = 24. A055096(n)^2 + a(n) is a square, as is A055096(n)^2 - a(n).

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

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

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

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

A057103 Triangle of congrua: T(n,k) = 4nk(n^2-k^2) with n>k>0 and starting at T(2,1) = 24. A055096(n)^2 + a(n) is a square, as is A055096(n)^2 - a(n).