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

Author

Henry Bottomley, Jan 31 2002

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

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 *)

Formula

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