A067756 Prime hypotenuses of Pythagorean triangles with a prime leg.

5, 13, 61, 181, 421, 1741, 1861, 2521, 3121, 5101, 8581, 9661, 16381, 19801, 36721, 60901, 71821, 83641, 100801, 106261, 135721, 161881, 163021, 199081, 205441, 218461, 273061, 282001, 337021, 388081, 431521, 491041, 531481, 539761, 552301

Offset: 1

Author

Henry Bottomley, Jan 31 2002

Keywords

• nonn

Comments

Apart from the first two terms, every term is congruent to 1 modulo 60 and is of the form 450k^2 +- 30k + 1 or 450k^2 +- 330k + 61 for some k.

Every term of the sequence after the second is a prime p congruent to 1 (mod 60), i.e., for n > 2, a(n) is a subsequence of A088955. The Pythagorean triple is {sqrt(2p-1), p-1, p}. - Lekraj Beedassy, Mar 12 2002

Primes p such that 2*p-1 is the square of a prime. - Robert Israel, Sep 16 2014

Primes p of the form ((q+1)/2)^2 + ((q-1)/2)^2, where q is a prime; then q belongs to A048161. - Thomas Ordowski, May 22 2015

The other (i.e., long) leg of the Pythagorean triangle is p-1. - Zak Seidov, Oct 30 2015

Examples

			For a(1)=5, the right triangle is 3, 4, 5 with 3 and 5 prime.
For a(10)=5101, the right triangle is 101, 5100, 5101 with 101 and 5101 prime.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000 (first 184 terms from Andreas Boe)
• H. Dubner and T. Forbes, Prime Pythagorean triangles, J. Integer Seqs., Vol. 4 (2001), #01.2.3.

Crossrefs

Contains every value of A051859.

Programs

Maple

N:= 10^8: # to get all terms <= N
Primes:= select(isprime,[$3..floor(sqrt(2*N-1))]):
f:= proc(p) local q; q:= (p^2+1)/2; if isprime(q) then q else NULL fi end proc:
map(f, Primes); # Robert Israel, Sep 16 2014

Mathematica

f[n_]:=((p-1)/2)^2+((p+1)/2)^2; lst={};Do[p=Prime[n];If[PrimeQ[f[p]],AppendTo[lst,f[p]]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 27 2009 *)

PARI

forprime(p=3,10^3,if(isprime(q=(p^2+1)/2),print1(q,", "))) \\ Derek Orr, Apr 30 2015

Formula

a(n) = (A048161(n)^2 + 1)/2 = A067755(n) + 1.