A165218 -id:A165218 - cp's OEIS frontend

A116886 Primes p that remain prime through at least 2 iterations of the function f(p) = p^2 + 4.

3, 17, 103, 137, 277, 313, 677, 743, 1117, 1627, 2003, 2143, 3407, 3677, 4483, 5087, 5903, 7177, 7333, 8087, 8093, 8147, 8537, 8573, 9293, 9473, 10177, 10477, 11173, 13807, 14897, 15107, 16657, 19753, 21563, 22307, 24113, 26113, 26417, 26633

Offset: 1

Giovanni Resta, Feb 27 2006

nonn

Numbers p with the property that p, q = p^2 + 4, and r = q^2 + 4 are all prime. - Zak Seidov, Sep 08 2009

a(n) = sqrt(A165218(n) - 4). - Zak Seidov, Sep 08 2009

			17 is prime, 17^2 + 4 = 293 is prime and 293^2 + 4 = 85853 is prime.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A062324, A116887, A116888, A116889, A045637, A062324, A165218.

Select[Prime[Range[2*7! ]],PrimeQ[ #^2+4]&&PrimeQ[(#^2+4)^2+4]&] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2010 *)
fQ[n_]:=AllTrue[Rest[NestList[#^2+4&,n,2]],PrimeQ]; Select[Prime[ Range[ 3000]],fQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 21 2014 *)

is(n)=my(q);isprime(p) && isprime(q=p^2+4) && isprime(q^2+4) \\ Charles R Greathouse IV, Nov 06 2013

Edited by N. J. A. Sloane, Sep 18 2009 at the suggestion of R. J. Mathar

A231120 Primes q of the form q = p^2 + 4 (p prime) such that r = q^2 + 4 and t = r^2 + 4 are also prime.

Original entry on oeis.org

25877573, 124835933, 277455653, 2210598293, 2970577013, 34314969053, 43115615453, 90247970573, 93738231893, 116836126973, 153183783773, 720880808213, 1818659924933, 2068485397733, 2258777543933, 3449960763653, 3645300477293, 4501767897173, 4670625512573, 5238481845533, 6812794277693

Offset: 1

Zak Seidov, Nov 04 2013

nonn

Positions of a(n) in A165218: 16, 29, 33, 74, 78, 105, 130, 333, 520, 547, 572, 716, 740, 820, 832, 865, 975.

The first primes such that t^2+4 is also prime are 93738231893, 2365771484804813, 4185535280578373, 4658429282719973, 7706774555568173, 7711174427503853, 25756066576859093.

Zak Seidov and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 699 terms from Seidov)

Subsequence of A165218.

forprime(p=2,1e7, if(isprime(q=p^2+4) && isprime(r=q^2+4) && isprime(r^2+4), print1(q", "))) \\ Charles R Greathouse IV, Nov 05 2013

A231235 Primes q of the form p^2 + 4 (p prime) such that r = q^2 + 4, s = r^2 + 4 and t = s^2 + 4 are all prime.

Original entry on oeis.org

93738231893, 2365771484804813, 4185535280578373, 4658429282719973, 7706774555568173, 7711174427503853, 25756066576859093, 65522912397466973, 80107252841869013, 105371595617867573, 130831138562692133, 174460360753737533, 201928181545454813, 204300010667474573

Offset: 1

Zak Seidov, Nov 06 2013

nonn

The next iteration is impossible: t^2 + 4 is divisible by 13.

Zak Seidov and Charles R Greathouse IV, Table of n, a(n) for n = 1..250 (first 100 terms from Zak Seidov)

Subsequence of A231120 and A165218.

Cf. A116889.

extnd[p_]:=NestList[#^2+4&,p,4]; #^2+4&/@Select[Prime[ Range[ 452*10^6]],AllTrue[Rest[extnd[#]],PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 06 2021 *)

Definition corrected by Harvey P. Dale, Jun 06 2021

cp's OEIS Frontend

A116886 Primes p that remain prime through at least 2 iterations of the function f(p) = p^2 + 4.

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A231120 Primes q of the form q = p^2 + 4 (p prime) such that r = q^2 + 4 and t = r^2 + 4 are also prime.

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A231235 Primes q of the form p^2 + 4 (p prime) such that r = q^2 + 4, s = r^2 + 4 and t = s^2 + 4 are all prime.

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