A122429 Primes p such that q = 4p^2 + 1, r = 4q^2 + 1 and s = 4r^2 + 1 are all primes.

13, 9833, 41647, 151607, 264757, 356123, 361223, 446863, 449093, 457813, 531383, 641057, 655927, 841697, 855947, 899263, 913687, 1052813, 1081757, 1379383, 1506493, 1575757, 1685087, 1821013, 1821377, 1981517, 2054233, 2142037

Offset

1,1

Comments

Next terms up to 400000th prime are 2286877, 2524157, 2595247, 2621737, 2931583, 3023437, 3425843, 3428567, 3538517, 3705187, 3777883, 3799717, 3875143, 3913727, 3973553, 4019833, 4167073, 4249523, 4488167, 4651873, 4822193, 4914937, 5054167, 5108293, 5140147, 5465303, 5520007, 5542003. - Zak Seidov, Jan 16 2009

Subsequence of A122424. - Pierre CAMI, Jul 21 2014

References

Clifford A. Pickover, A Passion for Mathematics, John Wiley & Sons, Inc., 2005, p.74.

Links

Pierre CAMI, Table of n, a(n) for n = 1..3636

Example

13 is there because 13, 677, 1833317 and 13444204889957 are prime.

Mathematica

Reap[Do[p=Prime[n]; q=4p^2+1; r=4q^2+1; s=4r^2+1; If[PrimeQ[{q, r, s}]=={True, True, True}, Sow[p]], {n, 15000}]][[2, 1]]
Select[Prime[Range[200000]], AllTrue[NestList[4#^2+1&, #, 3], PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 22 2015 *)

Prog

(PARI)
f(x)=4*x^2+1;
forprime(p=1, 10^8, if(isprime(f(p))&&isprime(f(f(p)))&&isprime(f(f(f(p)))), print1(p, ", "))) \\ Derek Orr, Jul 31 2014

Crossrefs

Cf. A052291, A005574, A001912, A122424.

Sequence in context: A020521 A208169 A205162 * A173506 A173840 A260457

Adjacent sequences: A122426 A122427 A122428 * A122430 A122431 A122432

Keyword

nonn

Author

Zak Seidov, Oct 20 2006

Extensions

More terms from Don Reble, Oct 24 2006

Edited by R. J. Mathar, Nov 02 2009

Status

approved