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Ordered product of the sides of primitive Pythagorean triangles divided by 60.
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1, 13, 34, 70, 203, 246, 259, 671, 1092, 1113, 1547, 2002, 3164, 3212, 3333, 3927, 4628, 5916, 7566, 8294, 9919, 10317, 10725, 17017, 17731, 17927, 21098, 24739, 26818, 29359, 30932, 34952, 40222, 40690, 49062, 50609, 51338, 53669, 59787
Hypotenuses of primitive Pythagorean triangles sorted on product of sides.
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5, 13, 17, 25, 29, 41, 37, 61, 65, 53, 85, 65, 113, 73, 101, 85, 89, 145, 97, 145, 109, 181, 125, 221, 149, 197, 137, 173, 265, 157, 185, 257, 169, 313, 185, 229, 193, 205, 365, 325, 205, 269, 233, 421, 221, 293, 241, 265, 401, 481, 317, 277, 545, 365, 485, 281
Scale factor by which primitive Pythagorean triangle {x= A088509(n), y= A088510(n), z= A088511(n)} needs be enlarged in order to circumscribe the smallest integral square having a side on the hypotenuse.
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37, 229, 409, 793, 1261, 2041, 1789, 4381, 5233, 4069, 8317, 6073, 14449, 7969, 12181, 9997, 11041, 23473, 14089, 24457, 17341, 36181, 20773, 53461, 29341, 44269, 28009, 38509, 76297, 35869, 44257, 74209, 42841, 105769, 50137, 65701, 53209
COMMENTS
Such an inscribed square has side x*y*z = A063011(n). Also the radius squared of the Conway circle of a primitive Pythagorean triangle, sorted on product of sides. - Frank M Jackson, Nov 04 2023
REFERENCES
J. D. E. Konhauser et al., Which Way Did The Bicycle Go?, Problem 21, "The Square on the Hypotenuse", pp. 7; 79-80, Dolciani Math. Exp. No. 18, MAA, 1996.
FORMULA
a(n) = x*y + z^2.
a(n) = s^2 + r^2, where s is the semiperimeter and r is the inradius of triangle (x, y, z).
MATHEMATICA
lst={}; k=25; Do[If[GCD[m, n]==1&&OddQ[m+n], AppendTo[lst, {2m*n(m^4-n^4), m^2(m+n)^2+n^2(m-n)^2}]], {m, 1, k}, {n, 1, m}]; lst=Sort@lst; Table[lst[[n]][[2]], {n, 1, 100}] (* Frank M Jackson, Nov 04 2023 *)
Short leg of primitive Pythagorean triangles sorted on product of sides.
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3, 5, 8, 7, 20, 9, 12, 11, 16, 28, 13, 33, 15, 48, 20, 36, 39, 17, 65, 24, 60, 19, 44, 21, 51, 28, 88, 52, 23, 85, 57, 32, 119, 25, 104, 60, 95, 84, 27, 36, 133, 69, 105, 29, 140, 68, 120, 96, 40, 31, 75, 115, 33, 76, 44, 160, 161, 136, 35, 207, 87, 48, 84, 204, 175, 37
Long leg of primitive Pythagorean triangles sorted on product of sides.
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4, 12, 15, 24, 21, 40, 35, 60, 63, 45, 84, 56, 112, 55, 99, 77, 80, 144, 72, 143, 91, 180, 117, 220, 140, 195, 105, 165, 264, 132, 176, 255, 120, 312, 153, 221, 168, 187, 364, 323, 156, 260, 208, 420, 171, 285, 209, 247, 399, 480, 308, 252, 544, 357, 483, 231
Ordered product of the terms in a primitive Pythagorean quadruple (with repetitions).
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12, 252, 288, 1008, 1188, 1872, 2052, 2100, 2448, 2772, 3300, 8400, 8448, 9108, 9828, 11628, 12768, 13500, 14688, 17100, 17388, 17388, 17472, 18900, 25500, 27900, 29568, 29568, 31968, 32292, 32508, 33408, 50388, 51612, 54000, 58212, 58812, 60372, 62100, 62832, 63072, 65472, 65892, 69300, 69972
COMMENTS
Every primitive Pythagorean quadruple (PPQ) generates a distinct Heronian triangle. This sequence is the area of such a triangle. If a, b, c, d form a PPQ where a^2 + b^2 + c^2 = d^2 it generates a primitive Heronian triangle whose three sides are b^2 + c^2, a^2 + c^2, a^2 + b^2. Its semiperimeter is d^2 and its area is a*b*c*d. It has an inradius and three exradii as a*b*c/d, b*c*d/a, a*c*d/b, a*b*d/c respectively.
a(n) == 0 mod 12.
A210484 is a subsequence because an integer Soddyian triangle has area m^2n^2(m+n)^2(m^2+mn+n^2) and semiperimeter (m^2+mn+n^2)^2 = m^2*n^2 + n^2(m+n)^2 + m^2(m+n)^2 where m >= n and GCD(m,n) = 1. This is a PPQ.
EXAMPLE
a(5)=1188 because the 5th occurrence of a PPQ sorted by the product of its term is (2, 6, 9, 11) and 1188 = 11*9*6*2.
MATHEMATICA
lst = {}; Do[lst=Join[lst, Select[PowersRepresentations[k^2, 3, 2], Times@@#!=0&&GCD@@#==1 &]], {k, 1, 100}]; lst1=Sort@(Table[{a, b, c}=lst[[n]]; a*b*c*Sqrt[a^2+b^2+c^2], {n, 1, Length@lst}]); lst1[[1;; 50]]
Ordered products of the perimeter and the sides of primitive Pythagorean triangles.
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720, 23400, 81600, 235200, 852600, 1305360, 1328400, 5314320, 8414280, 9434880, 16893240, 18498480, 33918720, 43995600, 45561600, 46652760, 57757440, 106226640, 108617760, 154736400, 155263680, 184041000, 235227600, 361712400, 417740400, 451760400, 471711240
COMMENTS
Considering the set of primitive Pythagorean triangles with sides (A, B, C), the sequence gives the values (A+B+C)*(A*B*C), in increasing order.
It is a challenge to find a pair of primitive Pythagorean triangles such that product of perimeter and the sides is equal.
EXAMPLE
a(1) = (3+4+5)*(3*4*5) = 720.
a(2) = (5+12+13)*(5*12*13) = 23400.
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