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 1-3 of 3 results.

A329148 Composite hypotenuses of primitive Pythagorean triangles (A120961) that are not circumdiameters of non-Pythagorean primitive Heronian triangles (A285579).

Original entry on oeis.org

1073, 1537, 1961, 3277, 4181, 5713, 7289, 7373, 8633, 9193, 9773, 10001, 10397, 11729, 13837, 14393, 14837, 14893, 15397, 16153, 16781, 17777, 17861, 19517, 20513, 20609, 20617, 20737, 21053, 21253, 21473, 21953, 22601, 23141, 23393
Offset: 1

Views

Author

Ray Chandler, Dec 04 2019

Keywords

Comments

Inspired by comments from Frank M Jackson in A285579.
All of the 1378 terms in the b-file are the product of two distinct Pythagorean primes.

Links

Crossrefs

Cf. A120961, A285579, A002144.

A146945 Hypotenuses of primitive Pythagorean triples which are not prime numbers and which are the hypotenuse of a unique triangle.

Original entry on oeis.org

25, 125, 169, 289, 625, 841, 1369, 1681, 2197, 2809, 3125, 3721, 4913, 5329, 7921, 9409, 10201, 11881, 12769, 15625, 18769, 22201, 24389, 24649, 28561, 29929, 32761, 37249, 38809, 50653, 52441, 54289, 58081, 66049, 68921, 72361, 76729, 78125
Offset: 1

Views

Author

John Harrison (harrison_uk_2000(AT)yahoo.co.uk), Apr 20 2009

Keywords

Comments

Each term is a prime power of the form p^e where p is in A002144 and e>1.
A proper subset of A120960 by eliminating A002144.
A proper subset of A120961 by eliminating A024409.
A proper subset of A008846 by eliminating A002144 and A024409.
A proper subset of A020882 by eliminating A002144, A024409 and duplicate entries.

Links

Crossrefs

Cf. A020882, A008846, A002144, A024409, A120960, A120961.

Programs

  • Mathematica
    lst1 = {1, 1}; lst2 = {}; Do[ If[ GCD[m, n] == 1, a = 2m*n; b = m^2 - n^2; c = m^2 + n^2; If[ !PrimeQ@c, AppendTo[lst1, c]]], {m, 3, 1000}, {n, If[OddQ@m, 2, 1], m - 1, 2}]; lst1 = Sort@ lst1; Do[ If[ lst1[[n - 1]] != lst1[[n]] && lst1[[n]] != lst1[[n + 1]], AppendTo[lst2, lst1[[n]]]], {n, 2, Length@ lst1 - 1}]; Take[lst2, 50] (* Robert G. Wilson v, May 02 2009 *)

Extensions

a(7) corrected by and a(17) and further terms from Robert G. Wilson v, May 02 2009
Minor edits to comments. - Ray Chandler, Nov 27 2019

A285579 Circumdiameters of non-Pythagorean primitive Heronian triangles (no repetitions).

Original entry on oeis.org

25, 65, 85, 125, 145, 169, 185, 205, 221, 265, 289, 305, 325, 365, 377, 425, 445, 481, 485, 493, 505, 533, 545, 565, 625, 629, 685, 689, 697, 725, 745, 785, 793, 841, 845, 865, 901, 905, 925, 949, 965, 985, 1025, 1037, 1105, 1145, 1157, 1165, 1189, 1205, 1241, 1261, 1285, 1313
Offset: 1

Views

Author

Frank M Jackson, Apr 22 2017

Keywords

Comments

All Pythagorean triangles have their hypotenuse equal to their circumdiameter. This sequence excludes Pythagorean triangles from the superset of primitive Heronian triangles (PHT's) and lists all circumdiameters with integer values excluding repetitions. It appears that this sequence is a subset of A120961, from which first differs at a(45). Missing terms start 1073, 1537, 1961, 3277, 4181, 5713, … (see A329148). The table of PHT's used to generate this sequence was obtained from lists generated by Sascha Kurz (see Link). The list contains a triple for every possible PHT with a maximum side length of 10000. The triples are in the form (a, b, c) where a >= b >= c and where a <= 10000.

Examples

			a(1) = 25 as the Heronian triangle with sides (20, 15, 7) is primitive, is non-Pythagorean, has an area of 42 and has a circumdiameter of 25. It is the first occurrence of an integral circumdiameter for a non-Pythagorean PHT.

Links

Crossrefs

Cf. A120961, A329148.

Programs

  • Mathematica
    lst1=ReadList["C:/primitive_heronian_triangles_1_10000.txt", {Number, Number, Number}]; lst = {}; Do[{a, b, c} = lst1[[n]]; s = (a + b + c)/2; A=Sqrt[s(s-a)(s-b)(s-c)]; If[IntegerQ[d=a*b*c/(2 A)]&&a!=d, AppendTo[lst, d]], {n, 1, Length[lst1]}];Union@lst (* For download of file of primitive Heronian triples see Link *)
Showing 1-3 of 3 results.

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