

A067756


Prime hypotenuses of Pythagorean triangles with a prime leg.


13



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
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OFFSET

0,1


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(2p1), p1, p}.  Lekraj Beedassy, Mar 12 2002
Primes p such that 2*p1 is the square of a prime.  Robert Israel, Sep 16 2014
The offset probably should be 1, but this sequence is exceptional and has offset 0. Please do not change it!  N. J. A. Sloane, Sep 30 2014
Primes p of the form ((q+1)/2)^2 + ((q1)/2)^2, where q is a prime; then q belongs to A048161.  Thomas Ordowski, May 22 2015
Other, larger, leg of Pythagorean triangle is p1.  Zak Seidov, Oct 30 2015


LINKS

Andreas Boe and Robert Israel, Table of n, a(n) for n = 0..10000 (0 to 183 from Andreas Boe).
H. Dubner and T. Forbes, Prime Pythagorean triangles, J. Integer Seqs., Vol. 4 (2001), #01.2.3.


FORMULA

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


EXAMPLE

For p(1)=5, the right triangle 3, 4, 5 is the smallest with 3 and 5 prime.
For p(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*N1))]):
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_]:=((p1)/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

Contains every value of A051859.
Sequence in context: A096639 A092773 A230444 * A284035 A051859 A151275
Adjacent sequences: A067753 A067754 A067755 * A067757 A067758 A067759


KEYWORD

nonn


AUTHOR

Henry Bottomley, Jan 31 2002


STATUS

approved



