cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A081752 Ordered product of the sides of primitive Pythagorean triangles divided by 60.

Original entry on oeis.org

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
Offset: 1

Views

Author

Lekraj Beedassy, Apr 08 2003

Keywords

Crossrefs

a(n)=A063011(n)/60. Subsequence of A057097.

Extensions

More terms from Ray Chandler, Oct 28 2003

A088511 Hypotenuses of primitive Pythagorean triangles sorted on product of sides.

Original entry on oeis.org

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
Offset: 1

Views

Author

Lekraj Beedassy, Nov 14 2003

Keywords

Crossrefs

Cf. A063011.
Cf. A057100.

Extensions

More terms from Ray Chandler, Nov 16 2003

A088544 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.

Original entry on oeis.org

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
Offset: 1

Views

Author

Lekraj Beedassy, Nov 17 2003

Keywords

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.

Links

Crossrefs

Cf. A088509, A088510, A088511, A063011.

Programs

  • 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 *)

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).

Extensions

More terms from Max Alekseyev, May 30 2009

A088509 Short leg of primitive Pythagorean triangles sorted on product of sides.

Original entry on oeis.org

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
Offset: 1

Views

Author

Lekraj Beedassy, Nov 14 2003

Keywords

Crossrefs

Cf. A063011.

Extensions

More terms from Ray Chandler, Nov 16 2003

A088510 Long leg of primitive Pythagorean triangles sorted on product of sides.

Original entry on oeis.org

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
Offset: 1

Views

Author

Lekraj Beedassy, Nov 14 2003

Keywords

Crossrefs

Cf. A063011.

Extensions

More terms from Ray Chandler, Nov 16 2003

A367737 Ordered product of the terms in a primitive Pythagorean quadruple (with repetitions).

Original entry on oeis.org

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
Offset: 1

Views

Author

Frank M Jackson, Nov 28 2023

Keywords

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.

Examples

			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.

Links

Crossrefs

Cf. A063011, A210484.

Programs

  • 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]]

A223857 Ordered products of the perimeter and the sides of primitive Pythagorean triangles.

Original entry on oeis.org

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
Offset: 1

Views

Author

Mihir Mathur, Apr 02 2013

Keywords

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.

Examples

			a(1) = (3+4+5)*(3*4*5) = 720.
a(2) = (5+12+13)*(5*12*13) = 23400.

Links

Crossrefs

Cf. A063011, A024364.

Extensions

Corrected and extended by Giovanni Resta, Apr 03 2013
Showing 1-7 of 7 results.

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