A122424 -id:A122424 - cp's OEIS frontend

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

Zak Seidov, Oct 20 2006

nonn

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

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

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

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

Cf. A052291, A005574, A001912, A122424.

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 *)

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

More terms from Don Reble, Oct 24 2006

Edited by R. J. Mathar, Nov 02 2009

A245746 Prime numbers P such that Q=4P^2+1, R=4Q^2+1, S=4R^2+1, T=4S^2+1 and Q, R, S, T are all prime numbers.

Original entry on oeis.org

9220303, 16079387, 17232253, 43606237, 66063373, 85403083, 97649917, 104719757, 159685553, 180467533, 197072563, 344777863, 492619373, 517774063, 647320727, 672712637, 715230127, 769494413, 790845563, 909545573, 944196137, 975302173, 1120585597, 1123182763

Offset: 1

Pierre CAMI, Jul 31 2014

nonn

Subsequence of A122429.

Pierre CAMI, Table of n, a(n) for n = 1..30

Cf. A052291, A122424, A122429.

pnQ[n_]:=Module[{q=4n^2+1,r,s,t},r=4q^2+1;s=4r^2+1;t=4s^2+1;AllTrue[ {q,r,s,t},PrimeQ]]; Select[Prime[Range[5678*10^4]],pnQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 08 2019 *)

cp's OEIS Frontend

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

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A245746 Prime numbers P such that Q=4P^2+1, R=4Q^2+1, S=4R^2+1, T=4S^2+1 and Q, R, S, T are all prime numbers.

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cp's OEIS Frontend

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

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A245746 Prime numbers P such that Q=4*P^2+1, R=4*Q^2+1, S=4*R^2+1, T=4*S^2+1 and Q, R, S, T are all prime numbers.

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A245746 Prime numbers P such that Q=4P^2+1, R=4Q^2+1, S=4R^2+1, T=4S^2+1 and Q, R, S, T are all prime numbers.