A089552 -id:A089552 - cp's OEIS frontend

A088319 Ordered hypotenuses of primitive Pythagorean triangles having legs that add up to a square.

41, 205, 389, 689, 1565, 1625, 1781, 3865, 4105, 4549, 5989, 7421, 9161, 9685, 10225, 10685, 13025, 17509, 17965, 18329, 21349, 21701, 25801, 33161, 33169, 33529, 36749, 38581, 39709, 49325, 51649, 52429, 52721, 56785, 57065, 67205, 70801

Offset: 1

Lekraj Beedassy, Nov 06 2003

nonn

			9161 is in the sequence because of the triple 5289^2 + 7480^2 = 9161^2 where we have 5289+7480=113^2.
Similarly, 205 belongs to the triple (133,156,205) and 133+156=17^2.

F. Rubin, "Squared" Pythagorean Triples, Solution to problem 2306, J. Recreational Mathematics, Vol. 29, No. 1, 1998, p. 73.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A088515, A088516, A088546, A089545-A089552, A089554-A089558.

terms = 1000; jmax = 100; kmax = 200;
Reap[Do[If[CoprimeQ[j, k], e = j^2 - j k + k^2/2; f = j k; If[e > f, Sow[e^2 + f^2]]], {j, 1, jmax}, {k, 2, kmax, 2}]][[2, 1]] // Union // Take[#, terms]& (* Jean-François Alcover, Mar 04 2020 *)

a(n)=e^2+f^2, where e>f, e=j^2 - jk + k^2/2 and f=jk for coprime pairs (j, k) with k even.

A088515 Values j of pairs (j,k) that generate A088319(n).

Original entry on oeis.org

1, 1, 5, 1, 7, 7, 1, 1, 9, 9, 3, 1, 11, 11, 3, 11, 1, 5, 13, 13, 1, 13, 3, 1, 15, 15, 5, 3, 15, 1, 5, 17, 17, 7, 17, 17, 1, 5, 7, 3, 19, 19, 19, 19, 5, 1, 19, 3, 21, 21, 7, 1, 21, 21, 7, 5, 23, 9, 23, 1, 23, 23, 3, 23, 7, 5, 9, 1, 23, 3, 25, 25, 7, 25, 25, 5, 25, 1, 7, 9, 27, 25, 27, 27, 5, 11, 1

Offset: 1

Lekraj Beedassy, Nov 14 2003

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A088319, A088516, A088546, A089545-A089552, A089554-A089558.

terms = 1000; jmax = 100; kmax = 200;
Reap[Do[If[CoprimeQ[j, k], e = j^2 - j k + k^2/2; f = j k; If[e > f, Sow[{e^2 + f^2, j, k}]]], {j, 1, jmax}, {k, 2, kmax, 2}]][[2, 1]] // Sort // #[[;; terms, 2]]& (* Jean-François Alcover, Mar 05 2020 *)

Corrected and extended by Ray Chandler, Nov 16 2003

A088516 Values k of pairs (j,k) that generate A088319(n).

Original entry on oeis.org

4, 6, 2, 8, 2, 4, 10, 12, 4, 2, 14, 14, 4, 6, 16, 2, 16, 18, 6, 4, 18, 2, 20, 20, 8, 4, 22, 22, 2, 22, 24, 6, 8, 24, 4, 2, 24, 26, 26, 26, 8, 6, 10, 4, 28, 26, 2, 28, 8, 10, 30, 28, 4, 2, 32, 32, 10, 32, 8, 30, 12, 6, 32, 4, 34, 34, 34, 32, 2, 34, 12, 8, 36, 14, 6, 36, 4, 34, 38, 38, 10, 2

Offset: 1

Lekraj Beedassy, Nov 14 2003

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A088319, A088515, A088546, A089545-A089552, A089554-A089558.

terms = 1000; jmax = 100; kmax = 200;
Reap[Do[If[CoprimeQ[j, k], e = j^2 - j k + k^2/2; f = j k; If[e > f, Sow[{e^2 + f^2, j, k}]]], {j, 1, jmax}, {k, 2, kmax, 2}]][[2, 1]] // Sort // #[[;; terms, 3]]& (* Jean-François Alcover, Mar 05 2020 *)

Corrected and extended by Ray Chandler, Nov 16 2003

A088546 Square root of sum of legs of primitive Pythagorean triangles having legs that add up to a square, sorted on hypotenuse.

Original entry on oeis.org

7, 17, 23, 31, 47, 41, 49, 71, 73, 79, 89, 97, 113, 103, 119, 119, 127, 137, 151, 161, 161, 167, 191, 199, 193, 217, 217, 233, 223, 241, 263, 271, 257, 239, 281, 287, 287, 313, 289, 329, 329, 343, 311, 353, 367, 337, 359, 383, 409, 391, 401, 391, 433, 439, 463

Offset: 1

Lekraj Beedassy, Nov 17 2003

nonn

Numbers whose square is the sum of the legs of primitive Pythagorean triangles with hypotenuse A088319(n).

			31 is in the sequence because it is associated with the primitive Pythagorean triangle (400,561,689) where 400+561=31^2.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A088319, A088515, A088516, A089545-A089552, A089554-A089558.

terms = 1000; jmax = 100; kmax = 200;
Reap[Do[If[CoprimeQ[j, k], e = j^2 - j k + k^2/2; f = j k; If[e > f, Sow[{e^2 + f^2, Abs[j^2 - k^2/2]}]]], {j, 1, jmax}, {k, 2, kmax, 2}]][[2, 1]] // Sort // #[[;; terms, 2]]& (* Jean-François Alcover, Mar 05 2020 *)

a(n) = abs(j^2 - k^2/2), where j=A088515(n), k=A088516(n).

a(n) = sqrt(A089552(n)).

More terms from Ray Chandler, Nov 16 2003

A089545 Values e of pairs (e,f) that generate A088319(n).

Original entry on oeis.org

5, 13, 17, 25, 37, 29, 41, 61, 53, 65, 65, 85, 85, 73, 89, 101, 113, 97, 109, 125, 145, 145, 149, 181, 137, 173, 157, 185, 197, 221, 193, 205, 185, 169, 229, 257, 265, 233, 205, 269, 241, 265, 221, 293, 277, 313, 325, 317, 305, 281, 289, 365, 365, 401, 337, 377

Offset: 1

Ray Chandler, Nov 16 2003

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A088319, A088515, A088516, A088546, A089546-A089552, A089554-A089558.

A089554 a(n)=(A088515(n)+1)/2.

Original entry on oeis.org

1, 1, 3, 1, 4, 4, 1, 1, 5, 5, 2, 1, 6, 6, 2, 6, 1, 3, 7, 7, 1, 7, 2, 1, 8, 8, 3, 2, 8, 1, 3, 9, 9, 4, 9, 9, 1, 3, 4, 2, 10, 10, 10, 10, 3, 1, 10, 2, 11, 11, 4, 1, 11, 11, 4, 3, 12, 5, 12, 1, 12, 12, 2, 12, 4, 3, 5, 1, 12, 2, 13, 13, 4, 13, 13, 3, 13, 1, 4, 5, 14, 13, 14, 14, 3, 6, 1, 2, 5, 4, 14, 6

Offset: 1

Ray Chandler, Nov 16 2003

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A088319, A088515, A088516, A088546, A089545-A089552, A089555-A089558.

A089558 a(n)=A089551(n)/2.

Original entry on oeis.org

2, 21, 35, 68, 161, 14, 155, 294, 306, 423, 483, 497, 902, 231, 984, 869, 776, 315, 1209, 1898, 1143, 1547, 2670, 1610, 1020, 3390, 2585, 3927, 2505, 2189, 4380, 5253, 3332, 84, 5474, 3791, 2892, 6695, 2093, 7449, 6764, 8607, 2945, 8246, 9590, 3731, 5453

Offset: 1

Ray Chandler, Nov 16 2003

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A088319, A088515, A088516, A088546, A089545-A089552, A089554-A089557.

A089546 Values f of pairs (e,f) that generate A088319(n).

Original entry on oeis.org

4, 6, 10, 8, 14, 28, 10, 12, 36, 18, 42, 14, 44, 66, 48, 22, 16, 90, 78, 52, 18, 26, 60, 20, 120, 60, 110, 66, 30, 22, 120, 102, 136, 168, 68, 34, 24, 130, 182, 78, 152, 114, 190, 76, 140, 26, 38, 84, 168, 210, 210, 28, 84, 42, 224, 160, 230, 288, 184, 30, 276, 138, 96, 92

Offset: 1

Ray Chandler, Nov 16 2003

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A088319, A088515, A088516, A088546, A089545-A089552, A089554-A089558.

A089551 Radius of inscribed circle within primitive Pythagorean triangles having legs that add up to a square, sorted on hypotenuse.

Original entry on oeis.org

4, 42, 70, 136, 322, 28, 310, 588, 612, 846, 966, 994, 1804, 462, 1968, 1738, 1552, 630, 2418, 3796, 2286, 3094, 5340, 3220, 2040, 6780, 5170, 7854, 5010, 4378, 8760, 10506, 6664, 168, 10948, 7582, 5784, 13390, 4186, 14898, 13528, 17214, 5890, 16492, 19180

Offset: 1

Ray Chandler, Nov 16 2003

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A088319, A088515, A088516, A088546, A089545-A089552, A089554-A089558.

A089547 Short leg of primitive Pythagorean triangles having legs that add up to a square, sorted on hypotenuse.

Original entry on oeis.org

9, 133, 189, 400, 1036, 57, 820, 1464, 1513, 2340, 2461, 2380, 5289, 973, 5617, 4444, 3616, 1309, 5797, 12921, 5220, 7540, 17880, 7240, 4369, 20760, 12549, 24420, 11820, 9724, 22849, 31621, 15729, 337, 31144, 17476, 12720, 37389, 8901, 41964, 34977

Offset: 1

Ray Chandler, Nov 16 2003

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A088319, A088515, A088516, A088546, A089545-A089552, A089554-A089558.

cp's OEIS Frontend

A088319 Ordered hypotenuses of primitive Pythagorean triangles having legs that add up to a square.

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A088515 Values j of pairs (j,k) that generate A088319(n).

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A088516 Values k of pairs (j,k) that generate A088319(n).

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A088546 Square root of sum of legs of primitive Pythagorean triangles having legs that add up to a square, sorted on hypotenuse.

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A089545 Values e of pairs (e,f) that generate A088319(n).

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A089554 a(n)=(A088515(n)+1)/2.

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A089558 a(n)=A089551(n)/2.

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A089546 Values f of pairs (e,f) that generate A088319(n).

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A089551 Radius of inscribed circle within primitive Pythagorean triangles having legs that add up to a square, sorted on hypotenuse.

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A089547 Short leg of primitive Pythagorean triangles having legs that add up to a square, sorted on hypotenuse.

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