Displaying 1-10 of 16 results found.
3, 7, 9, 13, 19, 15, 21, 31, 27, 33, 33, 43, 43, 37, 45, 51, 57, 49, 55, 63, 73, 73, 75, 91, 69, 87, 79, 93, 99, 111, 97, 103, 93, 85, 115, 129, 133, 117, 103, 135, 121, 133, 111, 147, 139, 157, 163, 159, 153, 141, 145, 183, 183, 201, 169, 189, 175, 153, 189, 211
Sum of legs of primitive Pythagorean triangles having legs that add up to a square, sorted on hypotenuse.
+10
18
49, 289, 529, 961, 2209, 1681, 2401, 5041, 5329, 6241, 7921, 9409, 12769, 10609, 14161, 14161, 16129, 18769, 22801, 25921, 25921, 27889, 36481, 39601, 37249, 47089, 47089, 54289, 49729, 58081, 69169, 73441, 66049, 57121, 78961, 82369
MATHEMATICA
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^2 - k^2/2)^2}]]], {j, 1, jmax}, {k, 2, kmax, 2}]][[2, 1]] // Sort // #[[;; terms, 2]]& (* Jean-François Alcover, Mar 05 2020 *)
Ordered hypotenuses of primitive Pythagorean triangles having legs that add up to a square.
+10
17
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
REFERENCES
F. Rubin, "Squared" Pythagorean Triples, Solution to problem 2306, J. Recreational Mathematics, Vol. 29, No. 1, 1998, p. 73.
FORMULA
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.
EXAMPLE
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.
MATHEMATICA
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 *)
Values j of pairs (j,k) that generate A088319(n).
+10
17
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
MATHEMATICA
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 *)
Values k of pairs (j,k) that generate A088319(n).
+10
17
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
MATHEMATICA
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 *)
Square root of sum of legs of primitive Pythagorean triangles having legs that add up to a square, sorted on hypotenuse.
+10
17
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
COMMENTS
Numbers whose square is the sum of the legs of primitive Pythagorean triangles with hypotenuse A088319(n).
EXAMPLE
31 is in the sequence because it is associated with the primitive Pythagorean triangle (400,561,689) where 400+561=31^2.
MATHEMATICA
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 *)
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
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
Values f of pairs (e,f) that generate A088319(n).
+10
3
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
Radius of inscribed circle within primitive Pythagorean triangles having legs that add up to a square, sorted on hypotenuse.
+10
3
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
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