A051858 -id:A051858 - cp's OEIS frontend

A067755 Even legs of Pythagorean triangles whose other leg and hypotenuse are both prime.

4, 12, 60, 180, 420, 1740, 1860, 2520, 3120, 5100, 8580, 9660, 16380, 19800, 36720, 60900, 71820, 83640, 100800, 106260, 135720, 161880, 163020, 199080, 205440, 218460, 273060, 282000, 337020, 388080, 431520, 491040, 531480, 539760, 552300

Offset: 1

Henry Bottomley, Jan 31 2002

nonn

Apart from the first two terms, every term is divisible by 60 and is of the form 450*k^2 +/- 30*k or 450*k^2 +/- 330*k + 60 for some k.

In such a triangle, this even leg is always the longer leg, and the hypotenuse = a(n) + 1. The Pythagorean triples are (A048161(n), a(n), A067756(n)), so, for a(2) = 12, the corresponding Pythagorean triple is (5, 12, 13). - Bernard Schott, Apr 12 2023

			4 is a term: in the right triangle (3, 4, 5), 3 and 5 are prime.
5100 is a term: in the right triangle (101, 5100, 5101), 101 and 5101 are prime.

Ray Chandler, Table of n, a(n) for n = 1..10000
H. Dubner and T. Forbes, Prime Pythagorean triangles, J. Integer Seqs., Vol. 4 (2001), #01.2.3.

Cf. A048161, A067756. Contains every value of A051858.

lst={}; Do[q=(Prime[n]^2+1)/2; If[PrimeQ[q], AppendTo[lst, (Prime[n]^2-1)/2]], {n, 200}]; lst (* Frank M Jackson, Nov 02 2013 *)

a(n) = (A048161(n)^2 - 1)/2 = A067756(n) - 1.

A051859 Values of C (the hypotenuse) of a Pythagorean triangle with A (the short leg) and C both prime and part of a twin prime.

Original entry on oeis.org

5, 13, 61, 181, 421, 5101, 60901, 135721, 161881, 163021, 218461, 595141, 1108561, 2574181, 2740141, 3248701, 3535141, 3723721, 3729181, 8197201, 13933921, 20218441, 23605321, 28569241, 33874681, 47248921, 68667481, 69372421

Offset: 1

Stuart M. Ellerstein (ellerstein(AT)aol.com), Dec 14 1999

nonn

All terms of the sequence must be part of a Pythagorean triple of the form (2u-1), 2u*(u-1), (2u^2 - 2u + 1). - Joshua Zucker, May 12 2006

Harvey P. Dale, Table of n, a(n) for n = 1..1000

See A051642 for the A's and A051858 for the B's.

Subset of A067756.

tppQ[{a_,b_,c_}]:=AllTrue[{a,c},PrimeQ]&&AnyTrue[a+{2,-2},PrimeQ] && AnyTrue[ c+{2,-2},PrimeQ]; Select[Table[{2n-1,2n(n-1),2n^2-2n+1},{n,2,10000}],tppQ][[All,3]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 27 2021 *)

A051858 + 1, or 2*A051891^2 - 2*A051891 + 1, or 2*A051892^2 + 2*A051892 + 1. - Joshua Zucker, May 12 2006

More terms from Joshua Zucker, May 12 2006

A051892 Values of e, the lesser key or generating number for Pythagorean triangles in which S (the odd short leg) and U (the hypotenuse) are twin primes.

Original entry on oeis.org

1, 2, 5, 9, 14, 50, 174, 260, 284, 285, 330, 545, 744, 1134, 1170, 1274, 1329, 1364, 1365, 2024, 2639, 3179, 3435, 3779, 4115, 4860, 5859, 5889, 6035, 6669, 8645, 8960, 8979, 9714, 9944, 10115, 10179, 15770, 16589, 16875, 19325, 20214, 20265, 21450

Offset: 1

Stuart M. Ellerstein (ellerstein(AT)aol.com), Dec 17 1999

nonn

Robert Simms, Deriving Pythagorean Triples (web archive)

Equals A051891 - 1. See A051642 for the S's, A051859 for the U's and A051858 for the T's (the even long leg).

More terms from Joshua Zucker, May 12 2006

A051891 Values of m, the main key or generating number for Pythagorean triangles in which S (the odd short leg) and U (the hypotenuse) are twin primes.

Original entry on oeis.org

2, 3, 6, 10, 15, 51, 175, 261, 285, 286, 331, 546, 745, 1135, 1171, 1275, 1330, 1365, 1366, 2025, 2640, 3180, 3436, 3780, 4116, 4861, 5860, 5890, 6036, 6670, 8646, 8961, 8980, 9715, 9945, 10116, 10180, 15771, 16590, 16876, 19326, 20215, 20266, 21451

Offset: 1

Stuart M. Ellerstein (ellerstein(AT)aol.com), Dec 17 1999

nonn

Robert Simms, Deriving Pythagorean Triples (web archive)

Cf. A051892. See A051642 for the S's, A051859 for the U's and A051858 for the T's (the even long leg).

a(n) = (A051642(n) + 1) / 2. - Sean A. Irvine, Oct 12 2021

More terms from Joshua Zucker, May 12 2006

cp's OEIS Frontend

A067755 Even legs of Pythagorean triangles whose other leg and hypotenuse are both prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A051859 Values of C (the hypotenuse) of a Pythagorean triangle with A (the short leg) and C both prime and part of a twin prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A051892 Values of e, the lesser key or generating number for Pythagorean triangles in which S (the odd short leg) and U (the hypotenuse) are twin primes.

Views

Author

Keywords

Links

Crossrefs

Extensions

A051891 Values of m, the main key or generating number for Pythagorean triangles in which S (the odd short leg) and U (the hypotenuse) are twin primes.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions