A135786 -id:A135786 - 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 *)

A050803 Cubes expressible as the sum of two nonzero squares in at least one way.

Original entry on oeis.org

8, 125, 512, 1000, 2197, 4913, 5832, 8000, 15625, 17576, 24389, 32768, 39304, 50653, 64000, 68921, 91125, 125000, 140608, 148877, 195112, 226981, 274625, 314432, 373248, 389017, 405224, 512000, 551368, 614125, 704969, 729000, 912673, 941192

Offset: 1

Patrick De Geest, Sep 15 1999

nonn

Root values equal terms from sequence A000404 'Sum of 2 nonzero squares'.
In other words, a(n)=(A000404(n))^3. - Artur Jasinski, Nov 29 2007
Obviously, if n and m are different members of this sequence, then n*m is also a member of this sequence. Additionally, if k^3 is a member of this sequence and k is not in A050804, then k^6 is also a member of this sequence. - Altug Alkan, May 11 2016

			551368 or 82^3 = 82^2 + 738^2 = 242^2 + 702^2.

Ian Stewart, "Game, Set and Math", Chapter 8 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124.

Index entries for sequences related to sums of squares

Cf. A000404, A050802, A050803, A135786, A135787, A135788.

a[n_]:=Module[{c=0},i=1; While[i^2Jayanta Basu, May 30 2013 *)
Select[Range[100]^3, Length[DeleteCases[PowersRepresentations[#, 2, 2], w_ /; MemberQ[w, 0]]] > 0 &] (* Michael De Vlieger, May 11 2016 *)

Edited by N. J. A. Sloane, May 15 2008 at the suggestion of R. J. Mathar

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

Henry Bottomley, Aug 02 2000

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

Cf. A073120, A256418.

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

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 *)

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

A135787 a(n) = A000404(n)^5.

Original entry on oeis.org

32, 3125, 32768, 100000, 371293, 1419857, 1889568, 3200000, 9765625, 11881376, 20511149, 33554432, 45435424, 69343957, 102400000, 115856201, 184528125, 312500000, 380204032, 418195493, 656356768, 844596301, 1160290625

Offset: 1

Artur Jasinski, Nov 29 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1400

Cf. A000404, A050803, A135784, A060803, A135786, A135788.

nMax = 100; p := Floor[Sqrt[nMax - 1]]; Union[Flatten[Table[a^2 + b^2, {a, p}, {b, a, Floor[Sqrt[nMax - a^2]]}]]]^5 (* G. C. Greubel, Nov 09 2016 *)

A135784 a(n) = A000404(n)^2.

Original entry on oeis.org

4, 25, 64, 100, 169, 289, 324, 400, 625, 676, 841, 1024, 1156, 1369, 1600, 1681, 2025, 2500, 2704, 2809, 3364, 3721, 4225, 4624, 5184, 5329, 5476, 6400, 6724, 7225, 7921, 8100, 9409, 9604, 10000, 10201, 10816, 11236, 11881, 12769, 13456, 13689, 14884

Offset: 1

Artur Jasinski, Nov 29 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1400

Cf. A000404, A060803, A135786, A135787, A135788.

nMax = 100; p := Floor[Sqrt[nMax - 1]]; Union[Flatten[Table[a^2 + b^2, {a, p}, {b, a, Floor[Sqrt[nMax - a^2]]}]]]^2 (* G. C. Greubel, Nov 09 2016 *)

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]

A135788 a(n) = A000404(n)^6.

Original entry on oeis.org

64, 15625, 262144, 1000000, 4826809, 24137569, 34012224, 64000000, 244140625, 308915776, 594823321, 1073741824, 1544804416, 2565726409, 4096000000, 4750104241, 8303765625, 15625000000, 19770609664, 22164361129, 38068692544

Offset: 1

Artur Jasinski, Nov 29 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1400

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

nMax = 100; p := Floor[Sqrt[nMax - 1]]; Union[Flatten[Table[a^2 + b^2, {a, p}, {b, a, Floor[Sqrt[nMax - a^2]]}]]]^6 (* G. C. Greubel, Nov 09 2016 *)

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]

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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A050803 Cubes expressible as the sum of two nonzero squares in at least one way.

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

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Maple

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A135787 a(n) = A000404(n)^5.

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A135784 a(n) = A000404(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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Maple

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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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Mathematica

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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Mathematica

A135788 a(n) = A000404(n)^6.

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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).