OFFSET
1,1
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
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.
EXAMPLE
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.
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 *)
PROG
(PARI) forprime(p=3, 10^3, if(isprime(q=(p^2+1)/2), print1(q, ", "))) \\ Derek Orr, Apr 30 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Henry Bottomley, Jan 31 2002
STATUS
approved