A060803 -id:A060803 - cp's OEIS frontend

A135786 a(n) = A000404(n)^4.

16, 625, 4096, 10000, 28561, 83521, 104976, 160000, 390625, 456976, 707281, 1048576, 1336336, 1874161, 2560000, 2825761, 4100625, 6250000, 7311616, 7890481, 11316496, 13845841, 17850625, 21381376, 26873856, 28398241, 29986576

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, 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]]}]]]^4 (* G. C. Greubel, Nov 09 2016 *)

Definition corrected by Zak Seidov, Aug 05 2009

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]

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.

A115245 Partial sums of A011764.

Original entry on oeis.org

3, 12, 93, 6654, 43053375, 1853020231905216, 3433683820292514337678080994497, 11790184577738583171520872861415952349498504106613519190091458

Offset: 0

Roger L. Bagula, Jun 07 2008

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..11

Cf. A011764, A060803.

Accumulate[3^(2^Range[0,10])] (* Harvey P. Dale, Oct 14 2012 *)

{a(n) = sum(k=0, n, 3^2^k)} \\ Seiichi Manyama, Sep 08 2019

cp's OEIS Frontend

A135786 a(n) = A000404(n)^4.

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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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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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Mathematica

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Crossrefs

A115245 Partial sums of A011764.

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Author

Keywords

Links

Crossrefs

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).