A116888 -id:A116888 - 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

A116887 Primes p that remain prime through at least 3 iterations of function f(p)=p^2+4.

Original entry on oeis.org

5087, 11173, 16657, 47017, 54503, 185243, 207643, 300413, 306167, 341813, 391387, 849047, 1348577, 1438223, 1502923, 1857407, 1909267, 2121737, 2161163, 2288773, 2610133, 2725157, 2744723, 2779097, 2874463, 2881327, 3079927, 3149077, 3154483, 3173683, 3194483

Offset: 1

Giovanni Resta, Feb 27 2006

nonn

			p=5087, f(p)= 25877573, f(f(p))= 669648784370333 and f(f(f(p)))= 448429494408664742387290530893 are all primes.

Zak Seidov, Table of n, a(n) for n = 1..2567 (all terms up to 10^9).

Cf. A062324, A116886, A116888, A116889.

Select[Prime[Range[8! ]],PrimeQ[ #^2+4]&&PrimeQ[(#^2+4)^2+4]&&PrimeQ[ ((#^2+4)^2+4)^2+4]&] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2010 *)

A116889 a(n) is the least prime p that remains prime through n iterations of function f(p)=p^2+4.

Original entry on oeis.org

2, 3, 3, 5087, 306167

Offset: 0

Giovanni Resta, Feb 27 2006

The sequence is finite, since it can be proved that if p, f(p), f(f(p)), f(f(f(p))) and f(f(f(f(p)))) are all primes, then the next iteration gives a multiple of 13, greater than 13, thus a(k) for k>=5 does not exist.

			a(0)=2 since f(2)=8 is not prime. a(1)=a(2)=3 since both f(3)=13 and f(f(3))=173 are primes.

Cf. A062324, A116886, A116887, A116888.

Typo in Example fixed by Zak Seidov, Nov 07 2013

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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A116887 Primes p that remain prime through at least 3 iterations of function f(p)=p^2+4.

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A116889 a(n) is the least prime p that remains prime through n iterations of function f(p)=p^2+4.

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